The Role of Short-term Consolidation in Memory Persistence

Timothy J. Ricker; Timothy J. Ricker

doi:10.3934/Neuroscience.2015.4.259

AIMS Neuroscience

2015, Volume 2, Issue 4: 259-279. doi: 10.3934/Neuroscience.2015.4.259

Previous Article Next Article

Research article Recurring Topics

The Role of Short-term Consolidation in Memory Persistence

Timothy J. Ricker ^,

The Memory and Decision Making Laboratory, Department of Psychology, College of Staten Island, City University of New York, Staten Island, NY, United States of America

Received: 18 August 2015 Accepted: 10 November 2015 Published: 12 November 2015

Short-term memory, often described as working memory, is one of the most fundamental information processing systems of the human brain. Short-term memory function is necessary for language, spatial navigation, problem solving, and many other daily activities. Given its importance to cognitive function, understanding the architecture of short-term memory is of crucial importance to understanding human behavior. Recent work from several laboratories investigating the entry of information into short-term memory has uncovered a dissociation between encoding processes, those that register information into short-term memory, and consolidation processes, those that solidify the representation within short-term memory. Here I describe the key differences between short-term encoding and consolidation and briefly review what is known about the short-term consolidation process itself. Cognitive function, plausible neural instantiation, and open questions are addressed.

Keywords:

Citation: Timothy J. Ricker. The Role of Short-term Consolidation in Memory Persistence[J]. AIMS Neuroscience, 2015, 2(4): 259-279. doi: 10.3934/Neuroscience.2015.4.259

Related Papers:

[1]	Boyu Wang . A splitting lattice Boltzmann scheme for (2+1)-dimensional soliton solutions of the Kadomtsev-Petviashvili equation. AIMS Mathematics, 2023, 8(11): 28071-28089. doi: 10.3934/math.20231436
[2]	Amna Mumtaz, Muhammad Shakeel, Abdul Manan, Marouan Kouki, Nehad Ali Shah . Bifurcation and chaos analysis of the Kadomtsev Petviashvili-modified equal width equation using a novel analytical method: describing ocean waves. AIMS Mathematics, 2025, 10(4): 9516-9538. doi: 10.3934/math.2025439
[3]	Jalil Manafian, Onur Alp Ilhan, Sizar Abid Mohammed . Forming localized waves of the nonlinearity of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model. AIMS Mathematics, 2020, 5(3): 2461-2483. doi: 10.3934/math.2020163
[4]	Cheng Chen . Hyperbolic function solutions of time-fractional Kadomtsev-Petviashvili equation with variable-coefficients. AIMS Mathematics, 2022, 7(6): 10378-10386. doi: 10.3934/math.2022578
[5]	Abeer S. Khalifa, Hamdy M. Ahmed, Niveen M. Badra, Jalil Manafian, Khaled H. Mahmoud, Kottakkaran Sooppy Nisar, Wafaa B. Rabie . Derivation of some solitary wave solutions for the (3+1)- dimensional pKP-BKP equation via the IME tanh function method. AIMS Mathematics, 2024, 9(10): 27704-27720. doi: 10.3934/math.20241345
[6]	Wafaa B. Rabie, Hamdy M. Ahmed, Taher A. Nofal, Soliman Alkhatib . Wave solutions for the (3+1)-dimensional fractional Boussinesq-KP-type equation using the modified extended direct algebraic method. AIMS Mathematics, 2024, 9(11): 31882-31897. doi: 10.3934/math.20241532
[7]	Gulnur Yel, Haci Mehmet Baskonus, Wei Gao . New dark-bright soliton in the shallow water wave model. AIMS Mathematics, 2020, 5(4): 4027-4044. doi: 10.3934/math.2020259
[8]	Jamilu Sabi'u, Sekson Sirisubtawee, Surattana Sungnul, Mustafa Inc . Wave dynamics for the new generalized (3+1)-D Painlevé-type nonlinear evolution equation using efficient techniques. AIMS Mathematics, 2024, 9(11): 32366-32398. doi: 10.3934/math.20241552
[9]	Xiaoli Wang, Lizhen Wang . Traveling wave solutions of conformable time fractional Burgers type equations. AIMS Mathematics, 2021, 6(7): 7266-7284. doi: 10.3934/math.2021426
[10]	Zhe Ji, Yifan Nie, Lingfei Li, Yingying Xie, Mancang Wang . Rational solutions of an extended (2+1)-dimensional Camassa-Holm- Kadomtsev-Petviashvili equation in liquid drop. AIMS Mathematics, 2023, 8(2): 3163-3184. doi: 10.3934/math.2023162

Abstract

1. Introduction

Nonlinear evolution equations (NLEEs) play an important part in the study of nonlinear science, particular in plasma physics, quantum field theory, nonlinear wave propagation and nonlinear optical fibers so that it attracted the attention of a large number of scholars. The extended auxiliary equation technique ^[1], the Bernoulli's equation approach ^[2], the Exp-function technique ^[3], the homotopy analysis technique ^[4], the homotopy perturbation technique ^[5], the improved $ tan(\phi/2) $-expansion technique (^[6,7]), the Hirota's bilinear technique ^{[8,9,10,11,12,13,14,15]}, the He's variational principle ^[16,17], the binary Darboux transformation ^[18], the Lie group analysis ^[19,20], the Bäcklund transformation method ^[21], optimal galerkin-homotopy asymptotic method applied ^[22], and the multiple rogue waves method (^[23,24]) have been proposed to solve NLEEs. By using these approaches, various exact solutions including soliton solution, lump solution, rogue wave solution, periodic solution, interaction solution, rational solution and high-order rational solution were obtained (^[25,26]).

In this paper, we mainly consider the following dynamical model, which can be used to describe some interesting (3+1)-dimensional waves of physics, namely, the generalized Kadomtsev-Petviashvili (gKP) equation ^[27]. That is

$\begin{equation} (\Psi_{t}+6\Psi\Psi_x+\Psi_{xxx})_x +a\Phi_{yy} = 0, \end{equation}$ $

(1.1)

and also above equation is integrable. Author of ^[28] introduced the modification of KP (mKP) equation ^[29] given

$\begin{equation} 4\Psi_{t}-6\Psi^2\Psi_x+\Psi_{xxx}+6\Psi_x\partial_{x}^{-1}\Psi_y +\Psi_{x}^{-1}\Phi_{yy} = 0. \end{equation}$ $

(1.2)

The generalized KP (gKP) equation has been researched by some scholars ^[30,31,32] in which is given as

$\begin{equation} (\Psi_{t}+\alpha\Psi_x +\beta\Psi\Psi_x+\gamma\Psi\Psi_{xxt})_x+\Psi_{yy} = 0. \end{equation}$ $

(1.3)

The another type of gKP equation is given in ^[33] as below

$\begin{equation} \Psi_{xxxy}+3(\Psi_x\Psi_y)_x+\Psi_{tx}+\Psi_{ty}+\Psi_{tz}-\Psi_{zz} = 0. \end{equation}$ $

(1.4)

We first present the bilinear form for Eq (1.4), by taking the following first-order logarithmic transformation

$\begin{equation} \Psi = 2(\ln f)_{x}, \end{equation}$ $

(1.5)

then, Eq (1.4) is turned into the bilinear form

$\begin{equation} \left(D_x^3 D_y+D_xD_t+D_yD_t+D_zD_t-D_z^2\right)\mathfrak{f}. \mathfrak{f} = 0, \end{equation}$ $

(1.6)

in which $ D_t, D_x, D_y $ and $ D_z $ are Hirota's bilinear frames. Cao ^[34] investigated the generalized B-type KP equation as follows

$\begin{equation} \Psi_{xxxy}+3(\Psi_x\Psi_y)_x-3\Psi_{xz}-\Psi_{ty} = 0. \end{equation}$ $

(1.7)

Guan et al. ^[35] derived a (3+1)-dimensional gKP equation in below form

$\begin{equation} \Psi_{xxxy}+3(\Psi_x \Psi_{y})_x+\alpha\Psi_{xxxz}+3\alpha(\Psi_x \Psi_{z})_x+\lambda_1\Psi_{xt}+\lambda_2\Phi_{yt}+\lambda_3\Phi_{zt}+ \omega_1\Phi_{xz} +\omega_2\Phi_{yz}+\omega_3\Phi_{zz} = 0, \end{equation}$ $

(1.8)

and some lump soliton solutions have been constructed using the Hirota bilinear method in ^[36]. Via transformation $ \Psi = 2(\ln f)_{x}, $ the bilinear form of equation (1.8) reads:

$\begin{equation} \left(D_x^3 D_y+\alpha D_x^3 D_z+\lambda_1D_xD_t+\lambda_2D_yD_t+\lambda_3D_zD_t+\omega_1D_xD_z+ \omega_2D_yD_z+\omega_3D_z^2\right)\mathfrak{f}. \mathfrak{f} = 0. \end{equation}$ $

(1.9)

Most classical test functions for solving NLPDEs by using the several particular functions can be constructed via Hirota bilinear technique. In other words, Hirota operator covers most of the classical hypothesis function method. For example, the fractional generalized CBS-BK equation ^[37], the generalized Bogoyavlensky-Konopelchenko equation ^[38], the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation ^[39], and the (2+1)-dimensional generalized variable-coefficient KP-Burgers-type equation ^[40]. Diverse kinds of studies on solve NLPDEs were perused via mighty authors in which some of them can be stated, for instance, multivariate rogue wave to some PDEs ^[41], interaction lump solutions the gKP equation ^[42], the (1+1)-dimensional coupled integrable dispersionless equations ^[43]. Therefore, we embark on the new research topic of constructing the analytic solutions of nonlinear PDEs by exploring the bilinear method. According to recent studies, we can obtain some of the new exact analytic solutions of nonlinear PDEs by way of constructing their corresponding bilinear differential equations in ^{[44,45,46,47]}. Here, we will study the multiple Exp-function method (MEFM) for determining the multiple soliton solutions (MSSs). The MEFM employed by some of powerful authors for various nonlinear equations have been surveyed in more studies in ^{[31,48,49,50]}. Authors of ^[51] utilized the reduced differential transform method for solving partial differential equations. Also, Yu and Sun ^[52] studied the dimensionally reduced generalized KP equations by help of Hirita bilinear method and obtained of lump solutions.

The outline of our paper is as follows: the multiple Exp-function scheme has been summarized in section 2. In sections 3, the KP equation, will investigate to finding 1-wave, 2-wave, and three-wave solutions. The Sections 4-6 devote to determined the periodic, cross-kink, and solitary wave solutions. Moreover, in Section 7, the modulation instability analysis is investigated. Finally in Section 8, the SIVP technique is considered with four cases for finding the solitary, bright, dark and singular wave solutions. A few of conclusions and outlook will be given in the final section.

2. Multiple Exp-function method

This method was summarized and improved for achieving the analytic solutions of NLPDEs:

Step 1. Assume a nonlinear PDE is given in general frame as follows

$\begin{equation} \mathcal{N}(x, y, t, \Psi , \Psi _{x}, \Psi _{y}, \Psi _{z}, \Psi _{t}, \Psi _{xx}, \Psi _{tt}, ...) = 0. \end{equation}$ $

(2.1)

Take the novel variables $ \xi _{i} = \xi _{i}(x, y, z, t), 1\leq i\leq n $, by differentiable frames:

$\begin{equation} \xi _{i, x} = \alpha _{i}\xi _{i}, \ \ \xi _{i, y} = \beta _{i}\xi _{i}, \ \ \xi _{i, z} = \gamma _{i}\xi _{i}, \ \ \xi _{i, t} = -\delta _{i}\xi _{i}, \ \ 1\leq i\leq n, \end{equation}$ $

(2.2)

where $ \alpha _{i}, \beta _{i}, \gamma _{i}, 1\leq i\leq n $, are unfound amounts. It noted that one can get as the following function

$\begin{equation} \xi _{i} = \varpi _{i}e^{\theta _{i}}, \ \ \theta _{i} = \alpha _{i}x+\beta _{i}y+\gamma _{i}z-\delta _{i}t, \ \ 1\leq i\leq n, \end{equation}$ $

(2.3)

where $ \varpi _{i}, 1\leq i\leq n $, unspecified amounts.

Step 2. Assuming the solution of the Eq (2.1) is function of variables $ \xi _{i}, 1\leq i\leq n $:

$\begin{equation} \Psi = \frac{\Delta (\xi _{1}, \xi _{2}, ..., \xi _{n})}{\Omega (\xi _{1}, \xi _{2}, ..., \xi _{n})}, \ \ \Delta = \sum\limits_{r, s = 1}^{n}\sum\limits_{i, j = 0}^{M}\Delta _{rs, ij}\xi _{r}^{i}\xi _{s}^{j}, \ \ \Omega = \sum\limits_{r, s = 1}^{n}\sum\limits_{i, j = 0}^{N}\Omega _{rs, ij}\xi _{r}^{i}\xi _{s}^{j}, \ \ \end{equation}$ $

(2.4)

in which $ \Delta _{rs, ij} $ and $ \Omega _{rs, ij} $ are amounts to be remained. Replacing Eq (2.4) into Eq (2.1) can be achieved the below form as:

$\begin{equation} \Psi = \frac{\Delta (\varpi _{1}e^{\alpha _{1}x+\beta _{1}y+\gamma _{1}z-\delta _{1}t}, ..., \varpi _{n}e^{\alpha _{n}x+\beta _{n}y+\gamma _{n}z-\delta _{n}t})}{\Omega (\varpi _{1}e^{\alpha _{1}x+\beta _{1}y+\gamma _{1}z-\delta _{1}t}, ..., \varpi _{n}e^{\alpha _{n}x+\beta _{n}y+\gamma _{n}z-\delta _{n}t})}, \end{equation}$ $

(2.5)

and also we have

$\begin{eqnarray} \Delta _{t} & = &\sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, t}, \ \ \Omega _{t} = \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, t}, \ \ \Delta _{x} = \sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, x}, \ \ \Omega _{x} = \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, x}, \ \ \Delta _{y} = \sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, y}, \ \\ \ \Delta _{z} & = &\sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, z}, \rm{ \ }\Omega _{y} = \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, y}, \ \ \Omega _{z} = \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, z}, \ \end{eqnarray}$ $

(2.6)

$ \ \ \ \Psi _{t} = \frac{\Omega \sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, t}-\Delta \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, t}}{\Omega ^{2}}, \ \ \Psi _{x} = \frac{\Omega \sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, x}-\Delta \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, x}}{\Omega ^{2}}, $

$ \Psi _{y} = \frac{\Omega \sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, y}-\Delta \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, y}}{\Omega ^{2}}, \ \ \Psi _{z} = \frac{ \Omega \sum\limits_{i = 1}^{n}\Delta _{\xi _{i}}\xi _{i, z}-\Delta \sum\limits_{i = 1}^{n}\Omega _{\xi _{i}}\xi _{i, z}}{\Omega ^{2}}. $

3. MSSs for the (3+1) gKP equation

3.1. Option I: 1-wave solution

The one-wave function of the solution will be reduced as below form

$\begin{equation} \Psi = \frac{2\Delta _{1}}{\Omega _{1}}, \ \ \Omega _{1} = 1+\rho _{1}+\rho _{2}\ e^{\alpha _{1}x+\beta _{1}y+\gamma _{1}z-\delta _{1}t}, \ \ \Delta _{1} = \sigma _{1}+\sigma _{2}\ e^{\alpha _{1}x+\beta _{1}y+\gamma _{1}z-\delta _{1}t}, \end{equation}$ $

(3.1)

in which $ \rho _{1}, \rho _{2}, \sigma _{1} $ and $ \sigma _{2} $ are unspecified amounts. Substituting (3.1) into Eq (1.8), the below cases will be concluded as:

Case I:

$\begin{equation} \alpha _{1} = \alpha _{1}, \ \ \beta _{1} = \beta _{1}, \ \ \rho _{1} = \rho _{1}, \ \ \rho _{2} = \ \ {\frac{\sigma _{{2}}\left( \rho _{{1}}+1\right) }{\sigma _{{1}}} \ \ } , \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{{2}} = \sigma _{2}, \ \ \delta _{{1} } = \delta _{1}, \ \ \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}. \end{equation}$ $

(3.2)

Case II:

$\begin{equation} \alpha _{1} = \alpha _{1}, \ \ \beta _{1} = - \ \ {\frac{\alpha \, {\alpha _{{1}}} \ \ ^{3}\gamma _{{1}}-\alpha _{{1}}\delta _{{1}}\lambda _{{1}}+\alpha _{{1} }\gamma _{{1}}\omega _{{1}}-\delta _{{1}}\gamma _{{1}}\lambda _{{3}}+{\gamma _{{1}}} ^{2}\omega _{{3}}} \ \ {{\alpha _{{1}}} ^{3}-\delta _{{1}}\lambda _{{2} }+\gamma _{{1}}\omega _{{2}}} }, \ \ \rho _{1} = -1, \ \ \rho _{2} = \rho _{2}, \ \ \end{equation}$ $

(3.3)

$ \sigma _{1} = \sigma _{1}, \ \ \sigma _{{2}} = \sigma _{2}, \ \ \delta _{{1} } = \delta _{1}, \ \ \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}. $

Case III:

$\begin{equation} \alpha _{1} = \ \ {\frac{\gamma _{{1}}\left( \alpha \, \delta _{{1}}\lambda _{{2} }-\alpha \, \gamma _{{1}}\omega _{{2}}-\delta _{{1}}\lambda _{{3}}+\gamma _{{1 }}\omega _{{3}}\right) }{\delta _{{1}}\lambda _{{1}}-\gamma _{{1}}\omega _{{1 }}} }, \ \ \beta _{1} = -\alpha \gamma _{1}, \ \ \rho _{1} = \rho _{1}, \ \ \rho _{2} = \rho _{2}, \ \ \end{equation}$ $

(3.4)

$ \sigma _{1} = \sigma _{1}, \ \ \sigma _{{2}} = \sigma _{2}, \ \ \delta _{{1} } = \delta _{1}, \ \ \ \ \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}. $

Case IV:

$\begin{eqnarray} \alpha _{1} & = &\vartheta , \ \ \beta _{1} = \beta _{1}, \ or\ \ \beta _{1} = -\alpha \gamma _{1}, \ \ \ \rho _{1} = -1, \ \ \rho _{2} = \rho _{2}, \ \ \sigma _{1} = \sigma _{1}, \ \ \\ \sigma _{{2}} & = &\sigma _{2}, \ \delta _{{1}} = \frac{\vartheta ^{3}+\gamma _{1}\omega _{2}}{\lambda _{2}}, \ \ \gamma_1 = \gamma_1, \ \ \gamma_2 = \gamma_2, \ \end{eqnarray}$ $

(3.5)

in which $ \vartheta $, solves the equation $ \lambda_1\vartheta ^4+(\gamma _1\lambda _3-\alpha \gamma _1\lambda_2) {\vartheta }^{3}+(\gamma _1\lambda _1\omega _2-\gamma_1\lambda _2\omega _1) \vartheta -\gamma _1 ^2\omega _3\lambda _2+\gamma_1^2\lambda _3\omega_2 = 0. $

For example, the 1-wave solution for Case III will be considered as

$\begin{equation} \Psi = 2\frac{\sigma _{1}+\sigma _{2}e^{ \ \ {\frac{\gamma _{{1}}\left( \alpha \, \delta _{{1}}\lambda _{{2}}-\alpha \, \gamma _{{1}}\omega _{{2}}-\delta _{{1 }}\lambda _{{3}}+\gamma _{{1}}\omega _{{3}}\right) }{\delta _{{1}}\lambda _{{ 1}}-\gamma _{{1}}\omega _{{1}}} \ \ }x-\alpha \gamma _{1}y+\gamma _{1}z-\delta _{1}t}}{1+\rho _{1}+\rho _{2}e^{ \ \ {\frac{\gamma _{{1}}\left( \alpha \, \delta _{ {1}}\lambda _{{2}}-\alpha \, \gamma _{{1}}\omega _{{2}}-\delta _{{1}}\lambda _{{3}}+\gamma _{{1}}\omega _{{3}}\right) }{\delta _{{1}}\lambda _{{1} }-\gamma _{{1}}\omega _{{1}}} \ \ }\ \ x-\alpha \gamma _{1}y+\gamma _{1}z-\delta _{1}t}}. \end{equation}$ $

(3.6)

3.2. Option II: 2-wave solutions

The two-wave function of the solution will be reduced as below form

$\begin{equation} \Psi = \frac{2\Delta _{2}}{\Omega _{2}}, \end{equation}$ $

(3.7)

$\begin{equation} \Omega _{2} = 1+\sigma _{1}e^{\alpha _{1}x+\beta _{1}y+\gamma _{1}z-\delta _{1}t}+\sigma _{2}e^{\alpha _{2}x+\beta _{2}y+\gamma _{2}z-\delta _{2}t}+\sigma _{1}\sigma _{2}\sigma _{12}e^{(\alpha _{1}+\alpha _{2})x+(\beta _{1}+\beta _{2})y+(\gamma _{1}+\gamma _{2})z-(\delta _{1}+\delta _{2})t}, \ \ \end{equation}$ $

(3.8)

$ \Delta _{2} = \rho _{1}e^{\alpha _{1}x+\beta _{1}y+\gamma _{1}z-\delta _{1}t}+\rho _{2}e^{\alpha _{2}x+\beta _{2}y+\gamma _{2}z-\delta _{2}t}+\rho _{1}\rho _{2}\rho _{12}e^{(\alpha _{1}+\alpha _{2})x+(\beta _{1}+\beta _{2})y+(\gamma _{1}+\gamma _{2})z-(\delta _{1}+\delta _{2})t}. $

Substituting (3.7) in terms of (3.8) into Eq (1.8), the below cases will be resulted as:

Case I:

$\begin{eqnarray} \alpha _{1} & = &0, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = \beta _{1}, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1}} = \ \ {\frac{\gamma _{{1}}\left( \beta _{{1}}\omega _{{2}}+\omega _{{3}}\gamma _{{1}}\right) }{\beta _{{1} }\lambda _{{2}}+\gamma _{{1}}\lambda _{{3}}} }, \ \ \\ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{ \ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = 0, \ \ \rho _{2} = \rho _{2}, \end{eqnarray}$ $

(3.9)

$ \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = 1. $

Case II:

$\begin{eqnarray} \alpha _{1} & = &\alpha _{1}, \ \ \alpha _{2} = 0, \ \ \beta _{1} = -\alpha \gamma _{1}, \ \ \beta _{2} = \beta _{2}, \ \ \delta _{{1}} = \ \ {\frac{\gamma _{{1}}\left( \alpha \, \gamma _{{1}}\omega _{{\ 2}}-\alpha _{{1}}\omega _{{1}}-\omega _{{3} }\gamma _{{1}}\right) }{\alpha \, \gamma _{{1}}\lambda _{{2}}-\alpha _{{1} }\lambda _{{1}}-\gamma _{{1}}\lambda _{{3}}} }, \\ \ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \beta _{{2}}\omega _{{2} }+\gamma _{{2}}\omega _{{3}}\right) }{\beta _{{2}}\lambda _{{2}}+\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = 0, \ \end{eqnarray}$ $

(3.10)

$ \ \rho _{2} = \rho _{2}, \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = 1. $

Case III:

$\begin{eqnarray} \alpha _{1} & = &\alpha _{1}, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = -\alpha \gamma _{1}, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1} } = \ \ {\frac{\gamma _{{1}}\left( \alpha \, \gamma _{{1}}\omega _{{\ 2}}-\alpha _{{ 1}}\omega _{{1}}-\omega _{{3}}\gamma _{{1}}\right) }{\alpha \, \gamma _{{1} }\lambda _{{2}}-\alpha _{{1}}\lambda _{{1}}-\gamma _{{1}}\lambda _{{3}}} }, \ \\ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{ {\ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = 0, \end{eqnarray}$ $

(3.11)

$ \ \ \rho _{2} = \rho _{2}, \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = 1. $

Case IV:

$\begin{eqnarray} \alpha _{1} & = &0, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = \beta _{1}, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1}} = \ \ {\frac{\gamma _{{1}}\left( \beta _{{1}}\omega _{{2}}+\omega _{{3}}\gamma _{{1}}\right) }{\beta _{{1} }\lambda _{{2}}+\gamma _{{1}}\lambda _{{3}}} }, \ \\ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{ {\ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = \rho _{1}, \ \ \rho _{2} = 0, \end{eqnarray}$ $

(3.12)

$ \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = 1. $

Case V:

$\begin{eqnarray} \alpha _{1} & = &\alpha _{1}, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = -\alpha \gamma _{1}, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1} } = \ \ {\frac{\gamma _{{1}}\left( \alpha \, \gamma _{{1}}\omega _{{\ 2}}-\alpha _{{ 1}}\omega _{{1}}-\omega _{{3}}\gamma _{{1}}\right) }{\alpha \, \gamma _{{1} }\lambda _{{2}}-\alpha _{{1}}\lambda _{{1}}-\gamma _{{1}}\lambda _{{3}}} }, \\ \ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{\ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{ \alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = \rho _{1}, \ \ \rho _{2} = 0, \end{eqnarray}$ $

(3.13)

$ \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = 1. $

Case VI:

$\begin{eqnarray} \alpha _{1} & = &\alpha _{1}, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = -{ \frac{\alpha \, \alpha _{{1}}\gamma _{{2}}} \ \ {\alpha _{{2}}} }, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1}} = \ \ {\frac{\alpha _{{1}}\gamma _{{2} }\left( \alpha \, \gamma _{{\ 2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1} }-\gamma _{{2}}\omega _{{3}}\right) }{\alpha _{{2}}\left( \alpha \, \gamma _{{ 2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2}}\lambda _{{3} }\right) }}, \ \\ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{ {\ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \gamma _{1}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = \rho _{1}, \ \ \rho _{2} = 0, \end{eqnarray}$ $

(3.14)

$ \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = \sigma _{12}. $

Case VII:

$\begin{eqnarray} \alpha _{1} & = &\alpha _{1}, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = -{ \frac{\alpha \, \alpha _{{1}}\gamma _{{2}}} \ \ {\alpha _{{2}}} }, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1}} = \ \ {\frac{\alpha _{{1}}\gamma _{{2} }\left( \alpha \, \gamma _{{\ 2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1} }-\gamma _{{2}}\omega _{{3}}\right) }{\alpha _{{2}}\left( \alpha \, \gamma _{{ 2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2}}\lambda _{{3} }\right) }}, \\ \ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{\ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{ \alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \ \ {\frac{\alpha _{{1}}\gamma _{{2}}} \ \ {\alpha _{{2} }}} \ \ , \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = \rho _{1}, \ \ \rho _{2} = \rho _{2}, \end{eqnarray}$ $

(3.15)

$ \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = 0, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = \sigma _{12}. $

Case VIII:

$\begin{eqnarray} \alpha _{1} & = &\frac{1}{2}\alpha _{2}, \ \ \alpha _{2} = \alpha _{2}, \ \ \beta _{1} = -\frac{1}{2}\, \alpha \, \gamma _{{2}}, \ \ \beta _{2} = -\alpha \gamma _{2}, \ \ \delta _{{1}} = \frac{1}{2}\, \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{ 1}}-\gamma _{{2}}\lambda _{{3}}} }, \ \\ \ \delta _{{2}} & = & \ \ {\frac{\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{ {\ 2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2} }\lambda _{{3}}} }, \gamma _{1} = \frac{1}{2}\gamma _{2}, \ \ \gamma _{2} = \gamma _{2}, \ \ \rho _{1} = \rho _{1}, \ \end{eqnarray}$ $

(3.16)

$ \ \rho _{2} = \rho _{2}, \ \ \rho _{12} = \rho _{12}, \ \ \sigma _{1} = \sigma _{1}, \ \ \sigma _{2} = \sigma _{2}, \ \ \sigma _{12} = 0. $

For instance, the 2-wave solution for Case I will be taken as

$\begin{equation} \Psi _{1} = \left. 2\, \rho _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{\ 2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2} }\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2} }\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{2}}-y\alpha \, \gamma _{{2}}+z\gamma _{{2}}} \ \ }\right/ \left( 1+\sigma _{{1}}{\mathrm{e}^{-{ \frac{t\gamma _{{1}}\left( \beta _{{1}}\omega _{{2}}+\omega _{{3}}\gamma _{{1 }}\right) }{\beta _{{1}}\lambda _{{2}}+\gamma _{{1}}\lambda _{{3}}} \ \ }+y\beta _{{1}}+z\gamma _{{1}}} \ \ }+\right. \end{equation}$ $

(3.17)

$ \left. \sigma _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{ 1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{2}}-y\alpha \, \gamma _{{2} }+z\gamma _{{2}}} \ \ }+\sigma _{{1}}\sigma _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{ 1}}\left( \beta _{{1}}\omega _{{2}}+\omega _{{3}}\gamma _{{1}}\right) }{ \beta _{{1}}\lambda _{{2}}+\gamma _{{1}}\lambda _{{3}}} \ \ }- \ \ {\frac{t\gamma _{{2} }\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1} }-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2} }-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{2} }+y\beta _{{1}}-y\alpha \, \gamma _{{2}}+z\gamma _{{1}}+z\gamma _{{2}}} } \right) . $

Also, the 2-wave solution for Case III will be considered as

$\begin{equation} \Psi _{2} = \left. 2\, \rho _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{\ 2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2} }\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2} }\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{2}}-y\alpha \, \gamma _{{2}}+z\gamma _{{2}}} \ \ }\right/ \left( 1+\sigma _{{1}}{\mathrm{e}^{-{ \frac{t\gamma _{{1}}\left( \alpha \, \gamma _{{1}}\omega _{{2}}-\alpha _{{1} }\omega _{{1}}-\omega _{{3}}\gamma _{{1}}\right) }{\alpha \, \gamma _{{1} }\lambda _{{2}}-\alpha _{{1}}\lambda _{{1}}-\gamma _{{1}}\lambda _{{3}}} \ \ } +x\alpha _{{1}}-y\alpha \, \gamma _{{1}}+z\gamma _{{1}}} \ \ }+\right. \end{equation}$ $

(3.18)

$ \left. \sigma _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{2}}\left( \alpha \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3} }\right) }{\alpha \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1} }-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{2}}-y\alpha \gamma _{{2}}+z\gamma _{{2}}} \ \ }+\sigma _{{1}}\sigma _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{1}}\left( \alpha \gamma _{{1}}\omega _{{2}}-\alpha _{{1}}\omega _{{1}}-\omega _{{3} }\gamma _{{1}}\right) }{\alpha \gamma _{{1}}\lambda _{{2}}-\alpha _{{1} }\lambda _{{1}}-\gamma _{{1}}\lambda _{{3}}} \ \ }- \ \ {\frac{t\gamma _{{2}}\left( \alpha \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2} }\omega _{{3}}\right) }{\alpha \gamma _{{2}}\lambda _{{2}}-\alpha _{{2} }\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{1}}+x\alpha _{{2} }-y\alpha \gamma _{{1}}-y\alpha \gamma _{{2}}+z\gamma _{{1}}+z\gamma _{{2}}} } \right) . $

And finally, the resulting two-wave solution for Case VIII will be read as

$\begin{equation} \Psi _{3}(x, y, z, t) = 2\left( \rho _{{1}}{\mathrm{e}^{-\frac{1}{2} \ \ {\frac{ t\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2 }}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+\frac{1}{2} x\alpha _{{2}}-\frac{1}{2}y\alpha \, \gamma _{{2}}+\frac{1}{2}z\gamma _{{2}}} \ \ } +\rho _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{2} }\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{ 2}}\lambda _{{3}}} \ \ }+x\alpha _{{2}}-y\alpha \, \gamma _{{2}}+z\gamma _{{2}}} \ \ } +\right. \end{equation}$ $

(3.19)

$ \left. \left. \rho _{{1}}\rho _{{2}}\rho _{{12}}{\ \mathrm{e}^{-\frac{3}{2}{ \frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2} }\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2} }\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+ \frac{3}{2}x\alpha _{{2}}-\frac{3}{2}y\alpha \, \gamma _{{2}}+\frac{3}{2} z\gamma _{{2}}} \ \ }\right) \right/ \left( 1+\sigma _{{1}}{\mathrm{e}^{-\frac{1}{ 2} \ \ {\frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2 }}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2} }\lambda _{{2}}-\alpha _{{2}}\lambda _{{1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+ \frac{1}{2}x\alpha _{{2}}-\frac{1}{2}y\alpha \gamma _{{2}}+\frac{1}{2} z\gamma _{{2}}} \ \ }\right. $

$ \left. +\sigma _{{2}}{\mathrm{e}^{- \ \ {\frac{t\gamma _{{2}}\left( \alpha \, \gamma _{{2}}\omega _{{2}}-\alpha _{{2}}\omega _{{1}}-\gamma _{{2}}\omega _{{3}}\right) }{\alpha \, \gamma _{{2}}\lambda _{{2}}-\alpha _{{2}}\lambda _{{ 1}}-\gamma _{{2}}\lambda _{{3}}} \ \ }+x\alpha _{{2}}-y\alpha \, \gamma _{{2} }+z\gamma _{{2}}} \ \ }\right) . $

3.3. Option III: Triple-wave solutions

The triple-wave function of the solution will be reduced as below form

$\begin{equation} \Psi = \frac{\Delta _{3}}{\Omega _{3}}, \end{equation}$ $

(3.20)

$\begin{equation} \Omega _{3} = 1+\rho _{1}e^{\Lambda _{1}}+\rho _{2}e^{\Lambda _{2}}+\rho _{3}e^{\Lambda _{3}}+\rho _{1}\rho _{2}\rho _{12}e^{\Lambda _{1}+\Lambda _{2}}+\rho _{1}\rho _{3}\rho _{13}e^{\Lambda _{1}+\Lambda _{3}}+\rho _{2}\rho _{3}\rho _{23}e^{\Lambda _{2}+\Lambda _{3}}+\rho _{1}\rho _{2}\rho _{3}\rho _{12}\rho _{13}\rho _{23}e^{\Lambda _{1}+\Lambda _{2}+\Lambda _{3}}, \ \ \end{equation}$ $

(3.21)

$ \Delta _{3} = 2\sigma _{1}e^{\Lambda _{1}}+2\sigma _{2}e^{\Lambda _{2}}+2\sigma _{1}\sigma _{2}\sigma _{12}e^{\Lambda _{1}+\Lambda _{2}}+2\sigma _{1}\sigma _{3}\sigma _{13}e^{\Lambda _{1}+\Lambda _{3}}+2\sigma _{2}\sigma _{3}\sigma _{23}e^{\Lambda _{2}+\Lambda _{3}}+ $

$ 2\sigma _{1}\sigma _{2}\sigma _{3}\sigma _{12}\sigma _{13}\sigma _{23}e^{\Lambda _{1}+\Lambda _{2}+\Lambda _{3}}, $

in which $ \Lambda _{i} = \alpha _{i}x+\beta _{i}y+\gamma _{i}x-\delta _{i}t, i = 1, 2, 3 $. Inserting (3.20) in terms of (3.21) into Eq (1.8), the below case will be reached:

$\begin{eqnarray} \alpha _{i} & = &\alpha _{i}, \ \ \gamma _{i} = \gamma _{i}, \ \ \sigma _{i} = \sigma _{i}, \ \ \rho _{1} = 0, \ \ \rho _{2} = \rho _{2}, \ \ \rho _{3} = \rho _{3}, \ \ \beta _{{i}} = -\alpha \, \gamma _{{i}}, \ \ \\ \delta _{{i}} & = & \ \ {\frac{\gamma _{{i}}\left( \alpha \, \gamma _{{i}}\omega _{{ \ 2}}-\alpha _{{i}}\omega _{{1}}-\omega _{{3}}\gamma _{{i}}\right) }{\alpha \, \gamma _{{i}}\lambda _{{2}}-\alpha _{{i}}\lambda _{{1}}-\gamma _{{i} }\lambda _{{3}}} }, \ \ i = 1, 2, 3, \ \ \rho _{ij} = \rho _{ij}, \ \ \sigma _{ij} = 1, \ \ i, j = 1, 2, 3, \ i\neq j. \end{eqnarray}$ $

(3.22)

Then, the solution is

$\begin{equation} \Psi _{1} = \frac{2\sigma _{1}e^{\Lambda _{1}}+2\sigma _{2}e^{\Lambda _{2}}+2\sigma _{1}\sigma _{2}e^{\Lambda _{1}+\Lambda _{2}}+2\sigma _{1}\sigma _{3}e^{\Lambda _{1}+\Lambda _{3}}+2\sigma _{2}\sigma _{3}e^{\Lambda _{2}+\Lambda _{3}}+2\sigma _{1}\sigma _{2}\sigma _{3}e^{\Lambda _{1}+\Lambda _{2}+\Lambda _{3}}} \ \ {1+\rho _{2}e^{\Lambda _{2}}+\rho _{3}e^{\Lambda _{3}}+\rho _{2}\rho _{3}\rho _{23}e^{\Lambda _{2}+\Lambda _{3}}} \ \ , \end{equation}$ $

(3.23)

in which $ \Lambda _{i} = \alpha _{i}x-\alpha \, \gamma _{{i}}y+\gamma _{i}x-{ \frac{\gamma _{{i}}\left(\alpha \, \gamma _{{i}}\omega _{{\ 2}}-\alpha _{{i} }\omega _{{1}}-\omega _{{3}}\gamma _{{i}}\right) }{\alpha \, \gamma _{{i} }\lambda _{{2}}-\alpha _{{i}}\lambda _{{1}}-\gamma _{{i}}\lambda _{{3}}} \ \ } t, i = 1, 2, 3 $ and $ \Psi = \Psi (x, y, z, t) $.

4. Novel periodic wave solutions

The triangular periodic waves for Eq (1.8) can be assumed as below:

$\begin{equation} \mathbf{f} = \exp (\tau _{1})+a_{16}\exp (-\tau _{1})+\cosh (\tau _{2})+\cos (\tau _{3})+a_{17}, \ \ \ \tau _{1} = \sum\limits_{i = 1}^{4}a_{i}x_{i}+a_{5}, \ \ \ \tau _{2} = \sum\limits_{i = 6}^{9}a_{i}x_{i-5}+a_{10}, \ \ \ \end{equation}$ $

(4.1)

$\begin{equation} \tau _{3} = \sum\limits_{i = 11}^{14}a_{i}x_{i-10}+a_{15}, \ \ (x_{1}, x_{2}, x_{3}, x_{4}) = (\mathbf{x}, t) = (x, y, z, t), \ \ \ \Psi (\mathbf{x} , t) = v_{0}+2\ln (\mathbf{f})_{x}, \end{equation}$ $

(4.2)

in which $ a_{i}, i = 1, ..., 17 $ are unfound values. Substituting (4.1) and (4.2) into Eq (1.8) the below consequences will be gained:

Case I:

$\begin{eqnarray} \mathbf{f} & = &{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{y\left( a_{{3}}a_{{14} }\omega _{{3}}-a_{{4}}a_{{12}}\omega _{{2}}-a_{{4}}a_{{13}}\omega _{{3} }\right) }{a_{{14}}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\cosh \left( a_{{9}}t-{ \frac{y\left( a_{{8}}a_{{14}}\omega _{{3}}-a_{{9}}a_{{12}}\omega _{{2}}-a_{{ \ 9}}a_{{13}}\omega _{{3}}\right) }{a_{{14}}\omega _{{2}}} \ \ }+a_{{8}}z+a_{{\ 10 }}\right) \\ &&+\cos \left( ta_{{14}}+ya_{{12}}+za_{{13}}+a_{{15}}\right) . \end{eqnarray}$ $

(4.3)

Appending (4.3) into (4.1) and (4.2), the soliton-periodic wave solution of Eq (1.8) as below will be achieved:

$\begin{equation} \Psi _{1} = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{ y\left( a_{{3}}a_{{14}}\omega _{{3}}-a_{{4}}a_{{12}}\omega _{{2}}-a_{{4}}a_{{ 13}}\omega _{{3}}\right) }{a_{{14}}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }}{ \mathbf{f}}. \end{equation}$ $

(4.4)

By selecting the suitable values of parameters including

$\begin{eqnarray*} a_{1} & = &1, a_{3} = 1.5, a_{4} = 2, a_{5} = 1.5, a_{8} = 2, a_{9} = 1.5, a_{10} = 1, \\ a_{12} & = &2, a_{13} = 2.5, a_{14} = 1, a_{15} = 3.2, \omega _{2} = 1.5, \omega _{3} = 1.2, \end{eqnarray*}$ $

the graphical display of soliton-periodic wave solution is offered in Figure 1 such as 3D plot and density plot.

Figure 1. The soliton-periodic solution (4.4) at (f1, f2) $ x = -2, y = -2 $, (f3, f4) $ x = 0, y = 0 $, and (f5, f6) $ x = 2, y = 2 $.

DownLoad: Full-Size Img PowerPoint

Case II:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+ya_{{2}}- \ \ {\frac { \left( {a_{{1}}} \ \ ^{3}a_{{\ 14}}-a_{{4}}a_{{13}}\omega_{{2}} \right) z}{a_{{14}}\omega_{{2}}} \ \ } +a_{ {5}}} \ \ }+\cosh \left( a_{{9}}t+ya_{{7}}+ \ \ {\frac {a_{{9}}a_{{13}}z}{a_{{14 } }}} \ \ +a_{{10}} \right) +\cos \left( ta_{{14}}+ya_{{12}}+za_{{13}}+a_{{\ 15}} \right). \end{equation}$ $

(4.5)

Plugging (4.5) into (4.1) and (4.2), obtain a soliton-periodic wave solution of Eq (1.8) as below case:

$\begin{equation} \Psi_2 = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+ya_{{2}}- \ \ {\frac { \left( {a_{{1}}} ^ {3}a_{{14}}-a_{{4}}a_{{13}}\omega_{{2}} \right) z}{a_{{14} }\omega_{{2} }}} \ \ +a_{{5}}} \ \ } }{\mathbf{f}} . \end{equation}$ $

(4.6)

Case III:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac {y \left( a_{{3}}\omega_{{3} }+a_{{4 }}\lambda_{{3}} \right) }{\omega_{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\cosh \left( a_{{9}}t- \ \ {\frac {y \left( a_{{8}}\omega_{{3}}+a_{{9}}\lambda_{ {3}} \right) }{\omega_{{2}}} \ \ }+a_{{8}}z+a_{{10}} \right) +\cos \left( - \ \ {\frac { a_{ {13}}\omega_{{3}}y}{\omega_{{2}}} \ \ }+za_{{13}}+a_{{15}} \right). \end{equation}$ $

(4.7)

Incorporating (4.7) into (4.1) and (4.2), the soliton-periodic wave solution of Eq (1.8) will be gained as below:

$\begin{equation} \Psi_3 = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac {y \left( a_{{3}}\omega_{{3} }+a_{{4}}\lambda_{{3}} \right) }{\omega_{{2}}} \ \ }+a_{{3} }z+a_{{5}}} \ \ } }{\mathbf{f}} . \end{equation}$ $

(4.8)

Case IV:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t- \ \ {\frac {y \left( a_{{3}}a_{{9}}\omega_{{3} }-a_{{4}} a_{{7}}\omega_{{2}}-a_{{4}}a_{{8}}\omega_{{3}} \right) }{a_{{9} }\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\cosh \left( a_{{9}}t+ya_{{7}}+a_{{8} }z+a_ {{10}} \right) +\cos \left( xa_{{11}}+a_{{15}} \right) . \end{equation}$ $

(4.9)

Plugging (4.9) into (4.1) and (4.2), the soliton-periodic wave solution of Eq (1.8) will be achieved as below:

$\begin{equation} \Psi_4 = v_{{0}}-\frac{2\sin \left( xa_{{11}}+a_{{15}} \right) a_{{11}} }{ \mathbf{f}}. \end{equation}$ $

(4.10)

Case V:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+ya_{{2}}+ \ \ {\frac {a_{{4}}a_{{8}}z}{a_{{9}}} \ \ } +a_{{5}} }}+\cosh \left( a_{{9}}t+ya_{{7}}+a_{{8}}z+a_{{10}} \right) +\cos \left( xa_{{11}}- \ \ {\frac { \left( {a_{{11}}} ^{2}\omega_{{3}}+\omega_{{\ 1} }\omega_{{2}} \right) a_{{11}}y}{{\omega_{{2}}} ^{2}}} \ \ + \ \ {\frac {{a_{{\ 11}}} \ \ ^{3}z}{\omega_{{2}}} \ \ }+a_{{15}} \right). \end{equation}$ $

(4.11)

Incorporating (4.11) into (4.1) and (4.2), we capture a soliton-periodic wave solution of Eq (1.8) as below:

$\begin{equation} \Psi_5 = v_{{0}}-\frac{2\sin \left( xa_{{11}}- \ \ {\frac { \left( {a_{{11}}} \ \ ^{2}\omega_{{3}}+ \omega_{{1}}\omega_{{2}} \right) a_{{11}}y}{{\omega_{{2}}} \ \ ^{2}}} \ \ +{\ \frac {{a_{{11}}} ^{3}z}{\omega_{{2}}} \ \ }+a_{{15}} \right) a_{{11}} }{ \mathbf{f}}. \end{equation}$ $

(4.12)

Case VI:

$\begin{eqnarray} \mathbf{f} & = &{\mathrm{e}^{a_{{4}}t+ \ \ {\frac{\omega _{{2}}\left( a_{{3}}a_{{9} }-a_{{4}}a_{{8}}\right) x}{a_{{9}}{a_{{11}}} ^{2}}} \ \ -\frac{1}{3}\, \ \ {\frac{ y\left( -2\, \alpha \, a_{{9}}{a_{{11}}} ^{6}\omega _{{2}}+2\, a_{{9}}{a_{{11}}} \ \ ^{6}\omega _{{3}}+3\, \alpha \, {a_{{3}}} ^{2}a_{{9}}{\omega _{{2}}} \ \ ^{3}-3\, \alpha \, a_{{3}}a_{{4}}a_{{8}}{\omega _{{2}}} ^{3}+2\, a_{{9}}{a_{{11}} }^{4}\omega _{{1}}\omega _{{2}}\right) }{{\omega _{{2}}} ^{3}\left( a_{{3}}a_{ {9}}-a_{{4}}a_{{8}}\right) }} \ \ +a_{{3}}z+a_{{5}}} }+ \\ &&\cosh \left( a_{{9}}t-\frac{1}{3} \ \ {\frac{y\Omega }{a_{{4}}{\omega _{{2}}} \ \ ^{3}\left( a_{{3}}a_{{9}}-a_{{4}}a_{{8}}\right) \left( {a_{{9}}} ^{2}{a_{{11}} }^{6}+{\omega _{{2}}} ^{2}\left( a_{{3}}a_{{9}}-a_{{4}}a_{{8}}\right) ^{2}\right) }}+a_{{8}}z+a_{{10}}\right) \\ &&+\cos \left( xa_{{11}}- \ \ {\frac{\alpha {a_{{11}}} ^{3}y}{\omega _{{2}}} \ \ }+{ \frac{{a_{{11}}} ^{3}z}{\omega _{{2}}} \ \ }+a_{{15}}\right) , \end{eqnarray}$ $

(4.13)

$\begin{eqnarray*} \Omega & = &3\, \alpha \, a_{{4}}a_{{8}}{\omega _{{2}}} ^{5}\left( a_{{3}}a_{{9} }-a_{{4}}a_{{8}}\right) ^{3}+\alpha \, {a_{{9}}} ^{2}{a_{{11}}} ^{6}{\omega _{{2 }}} ^{3}\left( a_{{3}}a_{{9}}+2\, a_{{4}}a_{{8}}\right) \left( a_{{3}}a_{{9} }-a_{{4}}a_{{8}}\right) \\ &&-{a_{{9}}} ^{2}{a_{{11}}} ^{4}{\omega _{{2}}} ^{2}\left( a_{{3}}a_{{9}}-a_{{4} }a_{{8}}\right) ^{2}\left( {a_{{11}}} ^{2}\omega _{{3}}+\omega _{{1}}\omega _{ {2}}\right) -{a_{{9}}} ^{2}{a_{{11}}} ^{12}\left( 3\, {a_{{4}}} ^{2}-{a_{{9}}} \ \ ^{2}\right) \left( \alpha \, \omega _{{2}}-\omega _{{3}}\right) . \end{eqnarray*}$ $

Appending (4.13) into (4.1) and (4.2), the soliton-periodic wave solution of Eq (1.8) will be obtained as below:

$\begin{eqnarray} \Psi _{5} & = &v_{{0}}+\frac{2}{\mathbf{f}}\left[ \ \ {\frac{\omega _{{2}}\left( a_{{3}}a_{{9}}-a_{{4}}a_{{8}}\right) }{a_{{9}}{a_{{11}}} ^{2}}{\mathrm{e}^{a_{ {4}}t+ \ \ {\frac{\omega _{{2}}\left( a_{{3}}a_{{9}}-a_{{4}}a_{{8}}\right) x}{a_{{ 9}}{a_{{11}}} ^{2}}} \ \ -\frac{1}{3}\, \ \ {\frac{y\left( -2\, \alpha \, a_{{9}}{a_{{11}} }^{6}\omega _{{2}}+2\, a_{{9}}{a_{{11}}} ^{6}\omega _{{3}}+3\, \alpha \, {a_{{3}} }^{2}a_{{9}}{\omega _{{2}}} ^{3}-3\, \alpha \, a_{{3}}a_{{4}}a_{{8}}{\omega _{{2 }}} ^{3}+2\, a_{{9}}{a_{{11}}} ^{4}\omega _{{1}}\omega _{{2}}\right) }{{\omega _{{2}}} ^{3}\left( a_{{3}}a_{{9}}-a_{{4}}a_{{8}}\right) }}+a_{{3}}z+a_{{5}}} \ \ }} \right. \\ &&\left. +\sin \left( -xa_{{11}}+ \ \ {\frac{\alpha \, {a_{{11}}} ^{3}y}{\omega _{{2 }}} \ \ }- \ \ {\frac{{a_{{11}}} ^{3}z}{\omega _{{2}}} \ \ }-a_{{15}}\right) a_{{11}}\right] . \end{eqnarray}$ $

(4.14)

By selecting suitable values of parameters including

$\begin{eqnarray*} \alpha & = &0.5, a_{3} = 1, a_{4} = 1.5, a_{5} = 2, a_{8} = 2, a_{9} = 1.5, a_{10} = 1, a_{11} = 2, a_{13} = 2.5, a_{14} = 1, \\ a_{15} & = &3.2, \omega _{1} = 1.5, \omega _{2} = 1.2, \omega _{3} = 1.5, \end{eqnarray*}$ $

the graphical display of soliton-periodic wave solution is offered in Figure 2 such as 3D chart and density chart.

Figure 2. The soliton-periodic solution (4.14) at (f1, f2) $ x = -2, y = -2 $, (f3, f4) $ x = 0, y = 0 $, and (f5, f6) $ x = 2, y = 2 $.

DownLoad: Full-Size Img PowerPoint

Case VII:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{ \ \ {\frac{\Omega \, a_{{9}}t}{{a_{{11}}} ^{3}}} \ \ +a_{{1}}x- \frac{1}{6}\, \ \ {\frac{y\left( -6\, {a_{{1}}} ^{4}{a_{{11}}} ^{5}\omega _{{3}}-3\, { a_{{1}}} ^{4}{a_{{11}}} ^{3}\omega _{{1}}\omega _{{2}}+2\, {a_{{11}}} ^{7}\omega _{{1}}\omega _{{2}}+6\, \Omega \, a_{{1}}a_{{8}}{a_{{11}}} ^{2}\omega _{{2} }\omega _{{3}}+3\, \Omega \, a_{{1}}a_{{8}}\omega _{{1}}{\omega _{{2}}} \ \ ^{2}\right) }{{a_{{11}}} ^{5}{\omega _{{2}}} ^{2}a_{{1}}} \ \ }+ \ \ {\frac{\left( -{a_{{ 1}}} ^{3}{a_{{11}}} ^{3}+\Omega \, a_{{8}}\omega _{{2}}\right) z}{{a_{{11}}} \ \ ^{3}\omega _{{2}}} \ \ }+a_{{5}}} \ \ }+ \end{equation}$ $

(4.15)

$ \cosh \left( a_{{9}}t+ya_{{7}}+a_{{8}}z+a_{{10}}\right) +\cos \left( xa_{{11} }-\frac{1}{2}\, \ \ {\frac{\left( 2\, {a_{{11}}} ^{2}\omega _{{3}}+\omega _{{1} }\omega _{{2}}\right) a_{{11}}y}{{\omega _{{2}}} ^{2}}} \ \ + \ \ {\frac{{a_{{11}}} ^{3}z }{\omega _{{2}}} \ \ }+a_{{15}}\right) , $

$ \Omega = \sqrt{-{a_{{1}}} ^{6}-2\, {a_{{1}}} ^{4}{a_{{11}}} ^{2}+{a_{{11}}} ^{6}}. $

Incorporating (4.15) into (4.1) and (4.2), the soliton-periodic wave solution of Eq (1.8) will be received as below:

$\begin{eqnarray} \Psi _{5} & = &v_{{0}}+\frac{2}{\mathbf{f}}\left[ a_{{1}}{\mathrm{e}^{ \ \ {\frac{ \Omega \, a_{{9}}t}{{a_{{11}}} ^{3}}} \ \ +a_{{1}}x-\frac{1}{6}\, \ \ {\frac{y\left( -6\, {a_{{1}}} ^{4}{a_{{11}}} ^{5}\omega _{{3}}-3\, {a_{{1}}} ^{4}{a_{{11}}} \ \ ^{3}\omega _{{1}}\omega _{{2}}+2\, {a_{{11}}} ^{7}\omega _{{1}}\omega _{{2} }+6\, \Omega \, a_{{1}}a_{{8}}{a_{{11}}} ^{2}\omega _{{2}}\omega _{{3} }+3\, \Omega \, a_{{1}}a_{{8}}\omega _{{1}}{\omega _{{2}}} ^{2}\right) }{{a_{{11 }}} ^{5}{\omega _{{2}}} ^{2}a_{{1}}} \ \ }+ \ \ {\frac{\left( -{a_{{1}}} ^{3}{a_{{11}}} \ \ ^{3}+\Omega \, a_{{8}}\omega _{{2}}\right) z}{{a_{{11}}} ^{3}\omega _{{2}}} \ \ } +a_{{5}}} \ \ }\right. \\ &&\left. -\sin \left( xa_{{11}}-\frac{1}{2}\, \ \ {\frac{\left( 2\, {a_{{11}}} \ \ ^{2}\omega _{{3}}+\omega _{{1}}\omega _{{2}}\right) a_{{11}}y}{{\omega _{{2}} }^{2}}} \ \ + \ \ {\frac{{a_{{11}}} ^{3}z}{\omega _{{2}}} \ \ }+a_{{15}}\right) a_{{11}} \right] . \end{eqnarray}$ $

(4.16)

By selecting the specific amounts of parameters including

$\begin{eqnarray*} \alpha & = &0.5, a_{1} = 1, a_{4} = a_{9} = \omega _{1} = \omega _{3} = 1.5, , a_{5} = 2, a_{7} = 1.5, a_{8} = 2, \\ a_{10} & = &1, a_{11} = 2, a_{13} = 2.5, a_{14} = 1, a_{15} = 3.2, \omega _{2} = 1.2, \end{eqnarray*}$ $

the graphical display of soliton-periodic wave solution is offered in Figure 3 such as 3D chart, density chart, and 2D chart and below cases:

$ (f3)\ y = 1, 2, 3, \ \ (f6)\ y = 1, 2, 3, \ and\ \ (f9)\ y = -1, -2, -3. $

Figure 3. The soliton-periodic solution (4.16) at (f1, f2) $ z = -2, t = -2 $, (f4, f5) $ z = 0, t = 0 $, and (f7, f8) $ z = 2, t = 2 $.

DownLoad: Full-Size Img PowerPoint

5. Novel cross-kink wave solutions

Three function containing exponential, hyperbolic, and triangular periodic waves for Eq (1.8) can be assumed as the following:

$\begin{equation} \mathbf{f} = \exp(\tau_1)+a_{16}\exp(-\tau_1)+\sinh(\tau_2)+\sin( \tau_3)+a_{17}, \ \ \ \tau_1 = \sum\limits_{i = 1}^{4}a_ix_i+a_5, \ \ \ \tau_2 = \sum\limits_{i = 6}^{9}a_ix_{i-5}+a_{10}, \end{equation}$ $

(5.1)

$\begin{equation} \tau_3 = \sum\limits_{i = 11}^{14}a_ix_{i-10}+a_{15}, \ \ (x_1, x_2, x_3, x_4) = (\mathbf{x} , t) = (x, y, z, t), \ \ \ \Psi(\mathbf{x}, t) = v_0+2\ln(\mathbf{f})_{x}, \end{equation}$ $

(5.2)

in which $ a_{i}, i = 1, ..., 17 $ are unfound values. Substituting (5.2) into Eq (1.8) the below consequences will be gained:

Case I:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{\left( a_{{3}}a_{{14} }\omega _{{3}}-a_{{4}}a_{{12}}\omega _{{2}}-a_{{4}}a_{{13}}\omega _{{3} }\right) y}{a_{{14}}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\sinh \left( a_{{9}}t- \ \ {\frac{\left( a_{{8}}a_{{14}}\omega _{{3}}-a_{{9}}a_{{12}}\omega _{{2}}-a_{{9 }}a_{{13}}\omega _{{3}}\right) y}{a_{{14}}\omega _{{2}}} \ \ }+a_{{8}}z+a_{{10} }\right) \end{equation}$ $

(5.3)

$ +\sin \left( ta_{{14}}+ya_{{12}}+za_{{13}}+a_{{15}}\right) . $

Substituting (5.3) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be gained as the following:

$\begin{equation} \Psi _{1} = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{\left( a_{{3}}a_{{14}}\omega _{{3}}-a_{{4}}a_{{12}}\omega _{{2}}-a_{{4}}a_{{13} }\omega _{{3}}\right) y}{a_{{14}}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }}{\mathbf{ f}}. \end{equation}$ $

(5.4)

By selecting the suitable values of parameters including

$\begin{eqnarray*} a_{1} & = &1, a_{3} = 3, a_{4} = 2, a_{5} = 1.5, a_{8} = 1.7, a_{9} = 1.5, a_{10} = 1.5, \\ a_{12} & = &2.5, a_{13} = 1.1, a_{14} = 2.1, a_{15} = 3.2, \omega _{2} = 1, \omega _{3} = 1.5, \end{eqnarray*}$ $

the graphical representation of cross-kink wave solution is offered in Figure 4 such 3D plot and density plot.

Figure 4. The cross-kink wave solution (5.4) at (f1, f2) $ x = 3, y = 2 $, (f3, f4) $ x = 0, y = 2 $, and (f5, f6) $ x = -3, y = 2 $.

DownLoad: Full-Size Img PowerPoint

Case II:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y- \ \ {\frac{\left( {a_{{1}}} \ \ ^{3}a_{{14}}-a_{{4}}a_{{13}}\omega _{{2}}\right) z}{a_{{14}}\omega _{{2}}} \ \ } +a_{{5}}} \ \ }+\sinh \left( a_{{9}}t+a_{{7}}y+ \ \ {\frac{a_{{9}}a_{{13}}z}{a_{{14}}} \ \ } +a_{{10}}\right) +\sin \left( ta_{{14}}+ya_{{12}}+za_{{13}}+a_{{15}}\right) . \end{equation}$ $

(5.5)

Putting (5.5) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be received as the following:

$\begin{equation} \Psi _{2} = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y-{ \frac{\left( {a_{{1}}} ^{3}a_{{14}}-a_{{4}}a_{{13}}\omega _{{2}}\right) z}{a_{ {14}}\omega _{{2}}} \ \ }+a_{{5}}} \ \ }}{\mathbf{f}}. \end{equation}$ $

(5.6)

By selecting the suitable values of parameters including

the graphical exhibition of cross-kink solution is offered in Figure 5 such as 3D chart and density chart.

Figure 5. The cross-kink wave solution (5.6) at (f1, f2) $ x = 3, y = 2 $, (f3, f4) $ x = 0, y = 2 $, and (f5, f6) $ x = -3, y = 2 $.

DownLoad: Full-Size Img PowerPoint

Case III:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac { \left( a_{{3}}\omega_{{3} }+a_{{4} }\lambda_{{3}} \right) y}{\omega_{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\sinh \left( a_{{9}}t- \ \ {\frac { \left( a_{{8}}\omega_{{3}}+a_{{9}}\lambda_{{\ 3}} \right) y}{\omega_{{2}}} \ \ }+a_{{8}}z+a_{{10}} \right) +\sin \left( - \ \ {\frac { a_{{13}}\omega_{{3}}y}{\omega_{{2}}} \ \ }+a_{{13}}z+a_{{15}} \right). \end{equation}$ $

(5.7)

Plugging (5.7) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be obtained as the following:

$\begin{equation} \Psi_3 = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac { \left( a_{{3}}\omega_{{3}} +a_{{4}}\lambda_{{3}} \right) y}{\omega_{{2}}} \ \ }+a_{{3} }z+a_{{5}}} \ \ } }{\mathbf{f}} . \end{equation}$ $

(5.8)

Case IV:

$\begin{eqnarray} \mathbf{f} & = &{\mathrm{e}^{- \ \ {\frac{a_{{9}}\left( {a_{{11}}} ^{2}\left( \omega _{{3}}{a_{{1}}} ^{2}-\omega _{{1}}\omega _{{2}}\right) \left( {a_{{1}}} \ \ ^{3}+a_{{3}}\omega _{{2}}\right) -{a_{{1}}} ^{5}\omega _{{1}}\omega _{{2} }\right) t}{{a_{{1}}} ^{2}a_{{7}}{a_{{11}}} ^{2}{\omega _{{2}}} ^{2}}} \ \ +a_{{1}}x- \ \ {\frac{a_{{3}}\left( \omega _{{3}}{a_{{1}}} ^{2}-\omega _{{1}}\omega _{{2} }\right) y}{{a_{{1}}} ^{2}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\sin \left( xa_{{ 11}}+a_{{15}}\right) \\ &&+\sinh \left( a_{{9}}t+a_{{7}}y- \ \ {\frac{\left( {a_{{1}}} ^{3}+a_{{3}}\omega _{{2}}\right) {a_{{1}}} ^{2}a_{{7}}{a_{{11}}} ^{2}\omega _{{2}}z}{{a_{{11}}} \ \ ^{2}\left( \omega _{{3}}{a_{{1}}} ^{2}-\omega _{{1}}\omega _{{2}}\right) \left( {a_{{1}}} ^{3}+a_{{3}}\omega _{{2}}\right) -{a_{{1}}} ^{5}\omega _{{1} }\omega _{{2}}} \ \ }+a_{{10}}\right) . \end{eqnarray}$ $

(5.9)

Inserting (5.9) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be gained as the following:

$\begin{equation} \Psi _{4} = v_{{0}}+2\frac{a_{{1}}{\mathrm{e}^{- \ \ {\frac{a_{{9}}\left( {a_{{11}}} \ \ ^{2}\left( \omega _{{3}}{a_{{1}}} ^{2}-\omega _{{1}}\omega _{{2}}\right) \left( {a_{{1}}} ^{3}+a_{{3}}\omega _{{2}}\right) -{a_{{1}}} ^{5}\omega _{{1} }\omega _{{2}}\right) t}{{a_{{1}}} ^{2}a_{{7}}{a_{{11}}} ^{2}{\omega _{{2}}} \ \ ^{2}}} \ \ +a_{{1}}x- \ \ {\frac{a_{{3}}\left( \omega _{{3}}{a_{{1}}} ^{2}-\omega _{{1} }\omega _{{2}}\right) y}{{a_{{1}}} ^{2}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ } +\cos \left( xa_{{11}}+a_{{15}}\right) a_{{11}}} \ \ {\mathbf{f}}. \end{equation}$ $

(5.10)

Case V:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{2}}y+ \ \ {\frac{a_{{4}}a_{{8}}z}{a_{{9}}} \ \ } +a_{{5}}} \ \ }+\sinh \left( a_{{9}}t+a_{{7}}y+a_{{8}}z+a_{{10}}\right) +\sin \left( xa_{{11}}- \ \ {\frac{a_{{11}}\left( {a_{{11}}} ^{2}\omega _{{3}}+\omega _{{ 1}}\omega _{{2}}\right) y}{{\omega _{{2}}} ^{2}}} \ \ + \ \ {\frac{{a_{{11}}} ^{3}z}{ \omega _{{2}}} \ \ }+a_{{15}}\right) . \end{equation}$ $

(5.11)

Substituting (5.11) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be received as the following:

$\begin{equation} \Psi _{5} = v_{{0}}+\frac{2\cos \left( xa_{{11}}- \ \ {\frac{a_{{11}}\left( {a_{{11} }}^{2}\omega _{{3}}+\omega _{{1}}\omega _{{2}}\right) y}{{\omega _{{2}}} ^{2}} }+ \ \ {\frac{{a_{{11}}} ^{3}z}{\omega _{{2}}} \ \ }+a_{{15}}\right) a_{{11}}} \ \ {\mathbf{f }}. \end{equation}$ $

(5.12)

By selecting the suitable values of parameters including

the graphical exhibition of cross-kink solution is offered in Figure 6 such as 3D chart and density chart.

Figure 6. The cross-kink wave solution (5.6) at (f1, f2) $ x = 3, y = 2 $, (f3, f4) $ x = 0, y = 2 $, and (f5, f6) $ x = -3, y = 2 $.

DownLoad: Full-Size Img PowerPoint

Case VI:

$\begin{eqnarray} \mathbf{f} & = &{\mathrm{e}^{ \ \ {\frac{\Omega \, a_{{9}}t}{{a_{{11}}} ^{3}}} \ \ +a_{{1} }x-\frac{1}{12} \ \ {\frac{\left( 3\, \Omega \, a_{{1}}a_{{8}}\omega _{{2}}\left( 4\, {a_{{11}}} ^{2}\omega _{{3}}+3\, \omega _{{1}}\omega _{{2}}\right) -12\, {a_{ {1}}} ^{4}{a_{{11}}} ^{5}\omega _{{3}}-{a_{{11}}} ^{3}\omega _{{1}}\omega _{{2} }\left( 9\, {a_{{1}}} ^{4}-2\, {a_{{11}}} ^{2}{a_{{1}}} ^{2}-2\, {a_{{11}}} \ \ ^{4}\right) \right) y}{{a_{{11}}} ^{5}{\omega _{{2}}} ^{2}a_{{1}}} \ \ } + \ \ {\frac{ \left( -{a_{{1}}} ^{3}{a_{{11}}} ^{3}+\Omega \, a_{{8}}\omega _{{2}}\right) z}{ \omega _{{2}}{a_{{11}}} ^{3}}} \ \ +a_{{5}}} \ \ } \\ &&\sinh \left( a_{{9}}t+a_7y+a_{{8}}z+a_{{10}}\right) +\sin \left( xa_{{ 11}} \! - \! \frac{1}{12} \ \ {\frac{a_{11}\left( 12{a_{{11}}} ^{2}\omega _{{3} } \! + \! 7\, \omega _1\omega _2\right) y}{{\omega _{{2}}} ^{2}}} \ \ + \ \ {\frac{a_{11}^3z}\omega _2} \! + \! a_{15}\right), \end{eqnarray}$ $

(5.13)

$ \Omega = \sqrt{-3\, {a_{{1}}} ^{6}-4\, {a_{{1}}} ^{4}{a_{{11}}} ^{2}+{a_{{11}}} ^{6} }. $

Putting (5.13) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be concluded as the following:

$\begin{eqnarray} \Psi _{5} & = &v_{{0}}+\frac{2}{\mathbf{f}}\left[ a_{{1}}{\mathrm{e}^{ \ \ {\frac{ \Omega \, a_{{9}}t}{{a_{{11}}} ^{3}}} \ \ +a_{{1}}x-\frac{1}{12} \ \ {\frac{\left( 3\, \Omega \, a_{{1}}a_{{8}}\omega _{{2}}\left( 4\, {a_{{11}}} ^{2}\omega _{{3} }+3\, \omega _{{1}}\omega _{{2}}\right) -12\, {a_{{1}}} ^{4}{a_{{11}}} \ \ ^{5}\omega _{{3}}-{a_{{11}}} ^{3}\omega _{{1}}\omega _{{2}}\left( 9\, {a_{{1}}} \ \ ^{4}-2\, {a_{{11}}} ^{2}{a_{{1}}} ^{2}-2\, {a_{{11}}} ^{4}\right) \right) y}{{a_{{ 11}}} ^{5}{\omega _{{2}}} ^{2}a_{{1}}} \ \ }+ \ \ {\frac{\left( -{a_{{1}}} ^{3}{a_{{11}}} \ \ ^{3}+\Omega \, a_{{8}}\omega _{{2}}\right) z}{\omega _{{2}}{a_{{11}}} ^{3}}} \ \ +a_{{5}}} \ \ }\right. \\ &&+\left. \cos \left( xa_{{11}}-\frac{1}{12} \ \ {\frac{a_{{11}}\left( 12\, {a_{{11} }}^{2}\omega _{{3}}+7\, \omega _{{1}}\omega _{{2}}\right) y}{{\omega _{{2}}} \ \ ^{2}}} \ \ + \ \ {\frac{{a_{{11}}} ^{3}z}{\omega _{{2}}} \ \ }+a_{{15}}\right) a_{{11}} \right] . \end{eqnarray}$ $

(5.14)

By selecting suitable values of parameters including

$\begin{eqnarray*} \alpha & = &0.5, a_{3} = 1, a_{4} = a_{9} = \omega _{1} = \omega _{3} = 1.5, a_{5} = 2, a_{8} = 2, \\ a_{10} & = &1, a_{11} = 2, a_{13} = 2.5, a_{14} = 1, a_{15} = 3.2, \omega _{2} = 1.2, \end{eqnarray*}$ $

the graphical representation of cross-kink solution is offered in Figure 7 such as 3D chart and density chart.

Figure 7. The cross-kink wave solution (5.14) at (f1, f2) $ x = -2, y = -2 $, (f3, f4) $ x = 0, y = 0 $, and (f5, f6) $ x = 2, y = 2 $.

DownLoad: Full-Size Img PowerPoint

Case VII:

$\begin{eqnarray} \mathbf{f} & = &{\mathrm{e}^{ \ \ {\frac{\Omega a_{{9}}t}{{a_{{11}}} ^{3}}} \ \ +a_{{1} }x-\frac{1}{6}{\ \frac{y\left( -6{a_{{1}}} ^{4}{a_{{11}}} ^{5}\omega _{{3} }-3\, {a_{{1}}} ^{4}{a_{{11}}} ^{3}\omega _{{1}}\omega _{{2}}+2\, {a_{{11}}} \ \ ^{7}\omega _{{1}}\omega _{{2}}+6\, \Omega a_{{1}}a_{{8}}{a_{{11}}} \ \ ^{2}\omega _{{2}}\omega _{{3}}+3\, \Omega \, a_{{1}}a_{{8}}\omega _{{1}}{\ \omega _{{2}}} ^{2}\right) }{{a_{{11}}} ^{5}{\omega _{{2}}} ^{2}a_{{1}}} \ \ }+{\ \frac{\left( -{a_{{1}}} ^{3}{a_{{11}}} ^{3}+\Omega a_{{8}}\omega _{{2} }\right) z}{{a_{{11}}} ^{3}\omega _{{2}}} \ \ }+a_{{5}}} \ \ }+ \\ &&\cosh \left( a_{{9}}t+ya_{{7}}+a_{{8}}z+a_{{10}}\right) + \cos \left( xa_{{\ 11}}-\frac{1}{2} \ \ {\frac{\left( 2{a_{{11}}} ^{2}\omega _{{3}}+\omega _{{1} }\omega _{{2}}\right) a_{{11}}y}{{\omega _{{2}}} ^{2}}} \ \ + \ \ {\frac{{a_{{11}}} ^{3}z }{\omega _{{2}}} \ \ }+a_{{15}}\right), \end{eqnarray}$ $

(5.15)

$ \Omega = \sqrt{-{a_{{1}}} ^{6}-2\, {a_{{1}}} ^{4}{a_{{11}}} ^{2}+{a_{{11}}} ^{6}}. $

Inserting (5.15) into (5.1) and (5.2), the cross-kink solution of Eq (1.8) will be gained as the following:

(5.16)

By selecting suitable values of parameters including

$ \alpha = 0.5, a_1 = 1, a_5 = 1.5, a_8 = 2, a_9 = 0.5, a_{10} = 1.5, a_{15} = 3.2, \omega_1 = 1.5, \omega_2 = 1.2, \omega_3 = 1.5, $

the graphical representation of cross-kink solution is offered in Figure 8 such as 3D chart and density chart.

Figure 8. The lump-periodic solution (5.16) at (f1, f2) $ z = -2 $, (f4, f5) $ z = 0 $, and (f7, f8) $ z = 2 $.

DownLoad: Full-Size Img PowerPoint

6. Novel solitary wave solutions

Three function containing exponential, hyperbolic, and triangular periodic waves for Eq (1.8) can be assumed as the following:

$\begin{equation} \textbf{f} = \exp (\tau _{1})+a_{16}\exp (-\tau _{1})+\tanh (\tau _{2})+\tan (\tau _{3})+a_{17}, \ \ \ \tau _{1} = \sum\limits_{i = 1}^{4}a_{i}x_{i}+a_{5}, \ \ \ \tau _{2} = \sum\limits_{i = 6}^{9}a_{i}x_{i-5}+a_{10}, \ \ \ \end{equation}$ $

(6.1)

(6.2)

in which $ a_{i}, i = 1, ..., 17 $ are unfound values. Substituting (6.2) into Eq (1.8) the below consequences will be gained:

Case I:

$\begin{equation} \textbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y- \ \ {\frac{\left( {a_{{1}}} \ \ ^{3}+a_{{4}}\lambda _{{2}}\right) z}{\omega _{{2}}} \ \ }+a_{{5}}} \ \ }+\tanh \left( ya_{{7}}+a_{{10}}\right) +\tan \left( ya_{{12}}+a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.3)

Substituting (6.3) into (6.1) and (6.2), we can capture a solitary wave solution of Eq (1.8) as the following:

$\begin{equation} \Psi _{1} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y-{ \frac{\left( {a_{{1}}} ^{3}+a_{{4}}\lambda _{{2}}\right) z}{\omega _{{2}}} \ \ } +a_{{5}}} \ \ }}{\mathbf{f}}. \end{equation}$ $

(6.4)

Case II:

$\begin{equation} \textbf{f} = {\mathrm{e}^{a_{{1}}x+a_{{2}}y- \ \ {\frac{{a_{{1}}} ^{3}z}{\omega _{{2} }}} \ \ +a_{{5}}} \ \ }+\tanh \left( ya_{{7}}+a_{{10}}\right) +\tan \left( ya_{{12} }+a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.5)

Inserting (6.5) into (6.1) and (6.2), we can capture a solitary wave solution of Eq (1.8) as below:

$\begin{equation} \Psi _{2} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{1}}x+a_{{2}}y- \ \ {\frac{{a_{{ 1}}} ^{3}z}{\omega _{{2}}} \ \ }+a_{{5}}} \ \ }}{\mathbf{f}}. \end{equation}$ $

(6.6)

Case III:

$\begin{equation} \textbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y- \ \ {\frac{\left( {a_{{1}}} \ \ ^{3}\omega _{{3}}+a_{{4}}\lambda _{{3}}\omega _{{2}}\right) z}{\omega _{{2} }\omega _{{3}}} \ \ }+a_{{5}}} \ \ }+\tanh \left( ta_{{9}}+ya_{{7}}- \ \ {\frac{za_{{9} }\lambda _{{3}}} \ \ {\omega _{{3}}} \ \ }+a_{{10}}\right) +\tan \left( ya_{{12}}+a_{{ 15}}\right) +a_{{17}}. \end{equation}$ $

(6.7)

Putting (6.7) into (6.1) and (6.2), the solitary wave solution of Eq (1.8) can be indicated as below:

$\begin{equation} \Psi _{3} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y-{ \frac{\left( {a_{{1}}} ^{3}\omega _{{3}}+a_{{4}}\lambda _{{3}}\omega _{{2} }\right) z}{\omega _{{2}}\omega _{{3}}} \ \ }+a_{{5}}} \ \ }}{\mathbf{f}}. \end{equation}$ $

(6.8)

Case IV:

$\begin{equation} \textbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y+za_{{3}}+a_{{5}}} \ \ }-\tanh \left( \ \ {\frac{ya_{{8}}\omega _{{3}}} \ \ {\omega _{{2}}} \ \ }-za_{{8}}-a_{{10} }\right) -\tan \left( \ \ {\frac{ya_{{13}}\omega _{{3}}} \ \ {\omega _{{2}}} \ \ }-a_{{13} }z-a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.9)

Putting (6.9) into (6.1) and (6.2), the solitary wave of Eq (1.8) can be stated as the following:

$\begin{equation} \Psi _{4} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2} }y+za_{{3}}+a_{{5}}} \ \ }}{\mathbf{f}}. \end{equation}$ $

(6.10)

By selecting the suitable values of parameters including

$\begin{eqnarray*} a_{1} & = &1, a_{2} = 1.5, a_{3} = 2, a_{4} = 2, a_{5} = 1.5, a_{8} = 2, a_{9} = 1.5, a_{10} = 1.5, \\ a_{13} & = &2, a_{13} = 1.1, a_{15} = 3.2, a_{17} = 2, \omega _{2} = -1.5, \end{eqnarray*}$ $

$ \omega _{3} = 1.2, \lambda _{2} = 1, \lambda _{3} = 1.5, x = -2, t = 2, $

with the following components

$\begin{eqnarray*} \alpha & = & \ \ {\frac{{a_{{1}}} ^{3}\omega _{{3}}-a_{{1}}\omega _{{1}}\omega _{{2} }-a_{{2}}{\omega _{{2}}} ^{2}-a_{{3}}\omega _{{2}}\omega _{{3}}+a_{{4} }\lambda _{{2}}\omega _{{3}}-a_{{4}}\lambda _{{3}}\omega _{{2}}} \ \ {\omega _{{2} }{a_{{1}}} ^{3}}} \ \ , \ \ \\ \lambda _{{1}} & = &- \ \ {\frac{{a_{{1}}} ^{3}a_{{2}}\omega _{{2}}+{a_{{1}}} ^{3}a_{{ 3}}\omega _{{3}}+a_{{2}}a_{{4}}\lambda _{{2}}\omega _{{2}}+a_{{3}}a_{{4} }\lambda _{{2}}\omega _{{3}}} \ \ {a_{{1}}a_{{4}}\omega _{{2}}} }, \end{eqnarray*}$ $

the graphical representation of rational solitary solution is offered in Figure 9 such as 3D chart and density chart.

Figure 9. The rational solitary wave solution (6.10) at (f1) $ z = 0 $, (f2) $ z = 1 $, and (f3) $ z = 3 $.

DownLoad: Full-Size Img PowerPoint

Case V:

$\begin{equation} \ f = {\mathrm{e}^{a_{{1}}x- \ \ {\frac{a_{{3}}\omega _{{3}}y}{\omega _{{2}}} \ \ }+za_{{3}}+a_{{5}}} \ \ }-\tanh \left( \ \ {\frac{ya_{{8}}\omega _{{3}}} \ \ {\omega _{{2}} }}-za_{{8}}-a_{{10}}\right) -\tan \left( \ \ {\frac{ya_{{13}}\omega _{{3}}} \ \ { \omega _{{2}}} \ \ }-a_{{13}}z-a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.11)

Incorporating (6.11) into (6.1) and (6.2), the solitary solution of Eq (1.8) will be gained as below:

$\begin{equation} \Psi _{5} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{1}}x- \ \ {\frac{a_{{3}}\omega _{{3}}y}{\omega _{{2}}} \ \ }+za_{{3}}+a_{{5}}} \ \ }}{\mathbf{f}}, \ \ \alpha = \ \ {\frac{{ a_{{1}}} ^{2}\omega _{{3}}-\omega _{{1}}\omega _{{2}}} \ \ {{a_{{1}}} ^{2}\omega _{{ 2}}} \ \ }. \end{equation}$ $

(6.12)

Case VI:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{\left( a_{{3}}\omega _{{3} }+a_{{4}}\lambda _{{3}}\right) y}{\omega _{{2}}} \ \ }+za_{{3}}+a_{{5}}} \ \ }+\tanh \left( ta_{{9}}- \ \ {\frac{y\left( a_{{8}}\omega _{{3}}+a_{{9}}\lambda _{{3} }\right) }{\omega _{{2}}} \ \ }+za_{{8}}+a_{{10}}\right) -\tan \left( {\ \frac{ ya_{{13}}\omega _{{3}}} \ \ {\omega _{{2}}} \ \ }-a_{{13}}z-a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.13)

Appending (6.13) into (6.1) and (6.2), the solitary solution of Eq (1.8) can be written as the following:

$\begin{equation} \Psi _{6} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{ \left( a_{{3}}\omega _{{\ 3}}+a_{{4}}\lambda _{{3}}\right) y}{\omega _{{2}}} \ \ } +za_{{3}}+a_{{5}}} \ \ }}{\mathbf{f}}, \ \ \alpha = \ \ {\frac{{a_{{1}}} ^{2}\omega _{{3} }-\omega _{{1}}\omega _{{2}}} \ \ {{a_{{1}}} ^{2}\omega _{{2}}} \ \ }. \end{equation}$ $

(6.14)

Case VII:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{a_{{3}}\omega _{{3}}y}{ \omega _{{2}}} \ \ }+za_{{3}}+a_{{5}}} \ \ }+\tanh \left( ta_{{9}}- \ \ {\frac{ya_{{8} }\omega _{{3}}} \ \ {\omega _{{2}}} \ \ }+za_{{8}}+a_{{10}}\right) -\tan \left( \ \ {\frac{ ya_{{13}}\omega _{{3}}} \ \ {\omega _{{2}}} \ \ }-a_{{13}}z-a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.15)

Appending (6.15) into (6.1) and (6.2), the solitary solution of Eq (1.8) will be obtained as below:

$\begin{equation} \Psi _{7} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{a_{{3 }}\omega _{{3}}y}{\omega _{{2}}} \ \ }+za_{{3}}+a_{{5}}} \ \ }}{\mathbf{f}}, \ \ \alpha = \ \ {\frac{{a_{{1}}} ^{2}\omega _{{3}}-\omega _{{1}}\omega _{{2}}} \ \ {{a_{{1}}} \ \ ^{2}\omega _{{2}}} }, \ \ \lambda _{{2}} = \ \ {\frac{\lambda _{{3}}\omega _{{2}}} \ \ { \omega _{{3}}} \ \ }. \end{equation}$ $

(6.16)

By choosing the specific amounts of parameters including

$\begin{eqnarray*} a_{1} & = &1, a_{2} = 1.5, a_{3} = 2, a_{4} = 2, a_{5} = 1.5, a_{8} = 2, a_{9} = 1.3, a_{10} = 1.5, \\ a_{13} & = &2, a_{13} = 2, a_{15} = 3.2, a_{17} = 2, \omega _{2} = -1.5, \\ \omega _{3} & = &1.2, \lambda _{2} = 1, \lambda _{3} = 1.5, x = -2, t = 2, \end{eqnarray*}$ $

the graphical representation of rational solitary solution is offered in Figure 10 such as 3D chart and density chart.

Figure 10. The rational solitary wave solution (6.16) at (f1) $ z = 0 $, (f2) $ z = 1 $, and (f3) $ z = 3 $.

DownLoad: Full-Size Img PowerPoint

Case VIII:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y- \ \ {\frac { \left( {a_{{1}}} \ \ ^{3}a_{{\ 14}}-a_{{4}}a_{{13}}\omega_{{2}} \right) z}{a_{{14}}\omega_{{2}}} \ \ } +a_{ {5}}} \ \ }+\tanh \left( ta_{{9}}+ya_{{7}}+ \ \ {\frac {za_{{9}}a_{{13}}} \ \ {a_{{14 } }}} \ \ +a_{{10}} \right) +\tan \left( ta_{{14}}+ya_{{12}}+a_{{13}}z+a_{{\ 15}} \right) +a_{{17}} . \end{equation}$ $

(6.17)

Incorporating (6.17) into (6.1) and (6.2), the solitary solution of Eq (1.8) will be received as below:

$\begin{equation} \Psi_8 = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y-{ \frac { \left( {a_{{1} }}^{3}a_{{14}}-a_{{4}}a_{{13}}\omega_{{2}} \right) z}{ a_{{14}}\omega_{ {2}}} \ \ }+a_{{5}}} \ \ } }{\mathbf{f}}, \ \ \alpha = \ \ {\frac {{a_{{1}}} \ \ ^{2}\omega_{{3}}-\omega_{{1}}\omega_{{2}}} \ \ {{a_ {{1}}} ^{2}\omega_{{2}}} }, \ \ \lambda_{{2}} = - \ \ {\frac {a_{{13}}\omega_{{2}}} \ \ {a_{{14}}} }, \ \ \lambda_{{3}} = -{ \frac {a_{{13}}\omega_{{3}}} \ \ {a_{{14}}} \ \ }. \end{equation}$ $

(6.18)

Case IX:

$\begin{eqnarray} \mathbf{f} & = &{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{\left( a_{{3}}a_{{14} }\omega _{{3}}-a_{{4}}a_{{12}}\omega _{{2}}-a_{{4}}a_{{13}}\omega _{{3} }\right) y}{a_{{14}}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ } \end{eqnarray}$ $

(6.19)

$\begin{eqnarray} &&+\tanh \left( ta_{{9}}- \ \ {\frac{y\left( a_{{8}}a_{{14}}\omega _{{3}}-a_{{9}}a_{{12}}\omega _{{2}}-a_{{ \ 9}}a_{{13}}\omega _{{3}}\right) }{a_{{14}}\omega _{{2}}} \ \ }+za_{{8}}+a_{{\ 10 }}\right) \end{eqnarray}$ $

(6.20)

$\begin{eqnarray} &&+\tan \left( ta_{{14}}+ya_{{12}}+za_{{13}}+a_{{15}}\right) +a_{{17}}. \end{eqnarray}$ $

(6.21)

Appending (6.21) into (6.1) and (6.2), the solitary solution of Eq (1.8) can be reached as below:

$\begin{equation} \Psi _{9} = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{\left( a_{{3}}a_{{14}}\omega _{{3}}-a_{{4}}a_{{12}}\omega _{{2}}-a_{{4}}a_{{13} }\omega _{{3}}\right) y}{a_{{14}}\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }}{\mathbf{ f}}, \ \alpha = \ \ {\frac{{a_{{1}}} ^{2}\omega _{{3}}-\omega _{{1}}\omega _{{2}}} \ \ {{ a_{{1}}} ^{2}\omega _{{2}}} }, \ \end{equation}$ $

(6.22)

$ \lambda _{{1}} = - \ \ {\frac{{a_{{1}}} ^{2}\left( a_{{12}}\omega _{{2}}+a_{{\ 13} }\omega _{{3}}\right) }{a_{{14}}\omega _{{2}}} }, \ \lambda _{{3}} = - \ \ {\frac{a_{{ 12}}\omega _{{2}}+a_{{13}}\omega _{{3}}} \ \ {a_{{\ 14}}} \ \ }. $

Case X:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac {a_{{3}}\omega_{{3}}y}{ \omega_{{2}} }}+a_{{3}}z+a_{{5}}} \ \ }+\tanh \left( - \ \ {\frac {ya_{{8}}\omega_{{3}} }{ \omega_{{2}}} \ \ }+za_{{8}}+a_{{10}} \right) +\tan \left( ta_{{14}}-{\ \frac { ya_{{13}}\omega_{{3}}} \ \ {\omega_{{2}}} \ \ }+za_{{13}}+a_{{15}} \right) +a_{{17}}. \end{equation}$ $

(6.23)

Inserting (6.21) into (6.1) and (6.2), the solitary solution of Eq (1.8) will be gained as below form:

$\begin{equation} \Psi_{10} = v_{{0}}+\frac{2\, a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac {a_{{ 3}}\omega_{{3}}y}{ \omega_{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ } }{\mathbf{f}}, \ \alpha = { \frac {{a_{{1}}} ^{2}\omega_{{3}}-\omega_{{1}}\omega_{{2}}} \ \ {{a_ {{1}}} \ \ ^{2}\omega_{{2}}} \ \ } , \ \lambda_{{2}} = \ \ {\frac {\lambda_{{3}}\omega_{{2}}} \ \ { \omega_{{3}}} \ \ }. \end{equation}$ $

(6.24)

By choosing the specific amounts of parameters including

$\begin{eqnarray*} a_{1} & = &1, a_{3} = 1.5, a_{4} = 2, a_{5} = 1.5, a_{7} = 2, a_{8} = 2, a_{9} = 2.1, \\ a_{10} & = &1.5, a_{13} = 2, a_{13} = 2, a_{14} = 2.5, a_{15} = 3.2, \\ a_{17} & = &2, \omega _{2} = -1.5, \omega _{3} = 1.2, \lambda _{3} = 1.5, x = -2, t = 2, \end{eqnarray*}$ $

the graphical representation of rational solitary solution is offered in Figure 11 such as 3D chart and density chart.

Figure 11. The rational solitary wave solution (6.24) at (f1) $ z = -2 $, (f2) $ z = 0 $, and (f3) $ z = 2 $.

DownLoad: Full-Size Img PowerPoint

Case XI:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y- \ \ {\frac{\left( {a_{{1}}} \ \ ^{3}a_{{\ 14}}-a_{{4}}a_{{13}}\omega _{{2}}\right) z}{a_{{14}}\omega _{{2}}} \ \ } +a_{{5}}} \ \ }+\tanh \left( \ \ {\frac{ta_{{14}}a_{{8}}} \ \ {a_{{13}}} \ \ }+ya_{{7}}+za_{{8} }+a_{{10}}\right) +\tan \left( ta_{{14}}+za_{{13}}+a_{{15}}\right) +a_{{17}}. \end{equation}$ $

(6.25)

Inserting (6.25) into (6.1) and (6.2), the solitary solution of Eq (1.8) can be stated as below case:

$\begin{equation} \Psi _{11} = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x+a_{{2}}y-{ \frac{\left( {a_{{1}}} ^{3}a_{{14}}-a_{{4}}a_{{13}}\omega _{{2}}\right) z}{a_{ {14}}\omega _{{2}}} \ \ }+a_{{5}}} \ \ }}{\mathbf{f}}, \ \alpha = - \ \ {\frac{{a_{{1}}} \ \ ^{3}a_{{2}}+a_{{1}}a_{{3}}\omega _{{1}}+a_{{2}}a_{{3}}\omega _{{2}}+{a_{{3}}} \ \ ^{2}\omega _{{3}}} \ \ {{a_{{1}}} ^{3}a_{{3}}} }, \end{equation}$ $

(6.26)

$\begin{eqnarray*} \lambda _{{1}} & = & \ \ {\frac{{a_{{1}}} ^{3}a_{{2}}a_{{12}}a_{{13}}-{a_{{1}}} \ \ ^{3}a_{{3}}{a_{{12}}} ^{2}+a_{{2}}a_{{3}}a_{{12}}a_{{13}}\omega _{{2}}+a_{{2} }a_{{3}}{a_{{13}}} ^{2}\omega _{{3}}+a_{{2}}a_{{3}}a_{{13}}a_{{14}}\lambda _{{ 3}}} \ \ {a_{{3}}a_{{12}}a_{{1}}a_{{14}}} \ \ } \\ &&+\frac{-{a_{{3}}} ^{2}{a_{{12}}} ^{2}\omega _{{2}}-{a_{{3}}} ^{2}a_{{12}}a_{{ 13}}\omega _{{3}}-{a_{{3}}} ^{2}a_{{12}}a_{{14}}\lambda _{{3}}} \ \ {a_{{3}}a_{{12} }a_{{1}}a_{{14}}} \ \ , \end{eqnarray*}$ $

$ \lambda _{{2}} = - \ \ {\frac{a_{{13}}\left( a_{{12}}\omega _{{2}}+a_{{13}}\omega _{ {3}}+a_{{14}}\lambda _{{3}}\right) }{a_{{12}}a_{{14}}} \ \ }. $

Case XII:

$\begin{equation} \mathbf{f} = {\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{a_{{3}}\omega _{{3}}y}{ \omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }+\tanh \left( ta_{{9}}+ya_{{7}}- \ \ {\frac{za_{ {7}}\omega _{{2}}} \ \ {\omega _{{3}}} \ \ }+a_{{10}}\right) +\tan \left( ta_{{14}}-{\ \frac{ya_{{13}}\omega _{{3}}} \ \ {\omega _{{2}}} \ \ }+za_{{13}}+a_{{15}}\right) +a_{{ 17}}. \end{equation}$ $

(6.27)

Inserting (6.27) into (6.1) and (6.2), the solitary solution of Eq (1.8) will be gained as below form:

$\begin{equation} \Psi _{12} = v_{{0}}+\frac{2a_{{1}}{\mathrm{e}^{a_{{4}}t+a_{{1}}x- \ \ {\frac{a_{{3} }\omega _{{3}}y}{\omega _{{2}}} \ \ }+a_{{3}}z+a_{{5}}} \ \ }}{\mathbf{f}}, \ \alpha = -{ \frac{{a_{{1}}} ^{3}a_{{2}}+a_{{1}}a_{{3}}\omega _{{1}}+a_{{2}}a_{{3}}\omega _{{2}}+{a_{{3}}} ^{2}\omega _{{3}}} \ \ {{a_{{1}}} ^{3}a_{{3}}} }, \end{equation}$ $

(6.28)

$ \lambda _{{2}} = - \ \ {\frac{a_{{13}}\left( a_{{12}}\omega _{{2}}+a_{{13}}\omega _{ {3}}+a_{{14}}\lambda _{{3}}\right) }{a_{{12}}a_{{14}}} \ \ }. $

7. Stability analysis of Eq (1.8)

In the current section, we will analyze the continuous modulational instability of the nonlinear generalized KP equation. In addition, the feasibility of the localized waves in the present system is certified by linear stability analysis. First, we search the perturbed solution for the giving Eq (1.8) of the form

$\begin{equation} \Psi(x, y, z, t) = \zeta+\delta\ \Theta, \end{equation}$ $

(7.1)

where $ \Theta = \Theta(x, y, z, t) $ and $ \zeta $ is a steady state solution. Inserting (7.1) into Eq (1.8), become

$\begin{equation} \delta\, \ \ {\frac {\partial ^{4}}{\partial {x}^{3}\partial y}}\Theta +\alpha\, \delta\, \ \ {\frac {\partial ^{4}}{\partial {x}^{3} \partial z}}\Theta +3\, {\delta}^{2} \left( \ \ {\frac { \partial ^{2}}{\partial {x}^{2}}} \ \ \Theta \right) {\ \frac {\partial }{\partial y}}\Theta + 3\, {\delta}^{2 } \left( { \frac {\partial }{\partial x}}\Theta \right) \ \ {\frac {\partial ^{2}}{\partial x\partial y}}\Theta +3\, \alpha\, {\delta}^{2} \left( \ \ {\frac {\partial ^{2}}{ \partial {x}^{2}}} \ \ \Theta \right) \ \ {\frac {\partial } {\partial z}}\Theta + \end{equation}$ $

(7.2)

$ 3\, \alpha\, {\delta}^{2} \left( \ \ {\frac {\partial }{\partial x}}\Theta \right) {\ \frac {\partial ^{2}}{\partial x\partial z}}\Theta + \lambda_{{1}}\delta\, \ \ {\frac {\partial ^{2}}{\partial t\partial x}}\Theta +\lambda_{{2}}\delta\, { \frac {\partial ^{2}}{ \partial t\partial y}}\Theta + \lambda_{{3}}\delta\, { \ \frac {\partial ^{2}}{\partial t\partial z}}\Theta + \omega_{{1}}\delta\, { \frac {\partial ^{2}}{\partial x\partial z}}\Theta +\omega_{{2}}\delta\, { \frac {\partial ^{2}}{ \partial y\partial z}}\Theta +\omega_{{3}}\delta\, {\ \frac {\partial ^{2}}{\partial {z}^{2}}} \ \ \Theta = 0, $

by linerization Eq (7.2), one gets

$\begin{equation} \delta \ \ {\frac {\partial ^{4}}{\partial {x}^{3}\partial y}}\Theta +\alpha\delta \ \ {\frac {\partial ^{4}}{\partial {x}^{3} \partial z}}\Theta +\lambda_{{1} }\delta \ \ {\frac { \partial ^{2}}{\partial t\partial x}}\Theta + \lambda _{{2} }\delta \ \ {\frac {\partial ^{2}}{\partial t\partial y}}\Theta +\lambda_{{3} }\delta \ \ {\frac {\partial ^{2}}{\partial t \partial z}}\Theta +\omega_{{1} }\delta \ \ {\frac { \partial ^{2}}{\partial x\partial z}}\Theta +\omega_ {{2} }\delta \ \ {\frac {\partial ^{2}}{\partial y\partial z}}\Theta +\omega_{{3} }\delta \ \ {\frac {\partial ^{2}}{\partial {z}^ {2}}} \ \ \Theta = 0. \end{equation}$ $

(7.3)

Theorem 7.1. Assume that the solution of Eq (7.3) has the following case as

$\begin{equation} \Theta(x, y, z, t) = \rho_1\ e^{i(Mx+Ny+Pz+Bt)}, \end{equation}$ $

(7.4)

in which $ M, N, P $ are the normalized wave numbers, by plugging (7.4) into Eq (7.3), separation the coefficients of $ e^{i(Mx+Ny+Pz+Wt)} $ one gets

$\begin{equation} B(M, N, P) = \ \ {\frac {{M}^{3}P\alpha+{M}^{3}N-MP\omega_{{1}}-NP\omega_{{2}}-{P} ^{2} \omega_{{3}}} \ \ {M\lambda_{{1}}+N\lambda_{{2}}+P\lambda_{{3}}} \ \ }. \end{equation}$ $

(7.5)

Proof. By putting (7.4) into (7.3), becomes

$\begin{equation} \delta\, \ \ {\frac {\partial ^{4}}{\partial {x}^{3}\partial y}}\overline{\Theta} +\alpha\, \delta\, \ \ {\frac {\partial ^{4}}{\partial {x}^{3} \partial z}} \overline{\Theta} +\lambda_{{1}}\delta\, \ \ {\frac { \partial ^{2}}{\partial t\partial x}}\overline{\Theta} + \lambda _{{2}}\delta\, \ \ {\frac {\partial ^{2} }{\partial t\partial y}}\overline{\Phi} +\lambda_{{3}}\delta\, \ \ {\frac { \partial ^{2}}{\partial t \partial z}}\overline{\Theta} +\omega_{{1}}\delta\, \ \ {\frac { \partial ^{2}}{\partial x\partial z}}\overline{\Theta} +\omega_ {{2} }\delta\, \ \ {\frac {\partial ^{2}}{\partial y\partial z}}\overline{\Theta} +\omega_{{3}}\delta\, \ \ {\frac {\partial ^{2}}{\partial {z}^ {2}}} \ \ \overline{ \Theta} \end{equation}$ $

(7.6)

$ = {\mathrm{e}^{i \left( Mx+Ny+Pz+Bt \right) }}\delta\, \rho_{{1}} \left( {M} ^{3}P\alpha+{M}^{3}N-MP\omega_{{1}}-MB\lambda_{{1}}-NP\omega_{{2}}-NB \lambda_{{2}}-{P}^{2}\omega_{{3}}-PB\lambda_{{3}} \right), $

in which $ \overline{\Theta} = \Theta(x, y, z, t) $. By solving and simplifying we can determine the function of $ B(M, N, P) $ as the following

$\begin{equation} B(M, N, P) = \ \ {\frac {{M}^{3}P\alpha+{M}^{3}N-MP\omega_{{1}}-NP\omega_{{2}}-{P} ^{2} \omega_{{3}}} \ \ {M\lambda_{{1}}+N\lambda_{{2}}+P\lambda_{{3}}} \ \ }. \end{equation}$ $

(7.7)

Accordingly, the considered solution was obtained. Thereupon the proof is perfect.

It is easy to notice that modulation stability occurs when $ M\lambda_{{1} }+N\lambda_{{2}}+P\lambda_{{3}}\neq0 $. So, the modulation stability achieve spectrum $ \Upsilon(B) $ will be as below form:

$\begin{equation} \Upsilon(B) = \ \ {\frac {{M}^{3}P\alpha+{M}^{3}N-MP\omega_{{1}}-NP\omega_{{2}}-{P} ^{2} \omega_{{3}}} \ \ {M\lambda_{{1}}+N\lambda_{{2}}+P\lambda_{{3}}} \ \ }. \end{equation}$ $

(7.8)

In Figures 12-14 can be discovered that while the sign of $ B(M, N, P) $ is positive for all quantity of $ M $. Furthermore, in Figure 12 can be observed if the $ B(M, N, P) $ is positive or negative for some quantities of $ M $. Finally, in Figure 13 and 14 can be perceived that while the sign of $ B(M, N, P) $ is positive for all quantity of $ M $.

Figure 12. The graphic representation $ \Upsilon(B) $ for the wave number $ M $ via considering the diverse quantities $ \lambda_1 = 1, \lambda_2 = 1.5, \lambda_3 = 2, \omega_1 = 2.2, \omega_2 = -2, \omega_3 = 3, \alpha = 0.5 $.

DownLoad: Full-Size Img PowerPoint

Figure 13. The graphic representation $ \Upsilon(B) $ for the wave number $ M $ via considering the diverse quantities $ \lambda_1 = 1, \lambda_2 = 1.5, \lambda_3 = 2, \omega_1 = 2.2, \omega_2 = -2, \omega_3 = 3, \alpha = 0.5 $.

DownLoad: Full-Size Img PowerPoint

Figure 14. The graphic representation $ \Upsilon(B) $ for the wave number $ M $ via considering the diverse quantities $ \lambda_1 = 1, \lambda_2 = 1.5, \lambda_3 = 2, \omega_1 = 2.2, \omega_2 = -2, \omega_3 = 3, \alpha = 0.5 $.

DownLoad: Full-Size Img PowerPoint

8. Usage of SIVP for Eq (1.8)

Via employing the wave alteration $ \xi = k(x+ay+bz-ct) $ in Eq (1.8) once can gain to the below ODE as

$\begin{equation} k^2(\alpha b+a)\Psi^{\prime \prime \prime \prime }+6k(\alpha b+a)\Psi^{\prime }\Psi^{\prime \prime }+(ab\omega_{{2}}-ac\lambda_{{2}}+{b} ^{2}\omega_{{3}}-bc\lambda_{{3}}+b \omega_{{1}}-c\lambda_{{1}} )\Psi^{\prime \prime } = 0, \end{equation}$ $

(8.1)

in which $ \Psi = \Psi(\xi) $ and $ \Psi^{\prime } = \frac{d\Psi}{d \xi} $. According to the SIVP ^[16,17] and by multiplying Eq (8.1) with $ \Psi^{\prime } $ and integrating once respect to $ \xi $, the following stationary integral will be arises

$\begin{equation} J = \int_{0}^{\infty}\left[A_1\left(\Psi^{\prime }\Psi^{\prime \prime \prime }- \frac{1}{2}(\Psi^{\prime \prime 2}\right)+\frac{1}{3}A_2(\Psi^{\prime 3}+ \frac{1}{2}A_3(\Psi^{\prime 2}\right]d\xi, \end{equation}$ $

(8.2)

in which

$ A_1 = k^2(\alpha b+a), \ \ \ A_2 = 6k(\alpha b+a), \ \ \ A_3 = ab\omega_{{2} }-ac\lambda_{{2}}+{b}^{2}\omega_{{3}}-bc\lambda_{{3}}+b \omega_{{1} }-c\lambda_{{1}}. $

8.1. Case I

We utilize the solitary wave function as the following

$\begin{equation} u(\xi ) = \delta \ sech(\mu \xi ). \end{equation}$ $

(8.3)

Hence, the stationary integral transforms to

$\begin{eqnarray} J & = &\frac{1}{30}{k}^{2}{\delta }^{2}\mu \, -21\, \alpha \, b{k}^{2}{\mu } ^{2}-21\, a{k}^{2}{\mu }^{2}-8\, \alpha \, b\delta \, k\mu -8\, a\delta \, k\mu +5\, ab\omega _{{2}} \\ &&-5\, ac\lambda _{{2}}+5\, {b}^{2}\omega _{{3}}-5\, bc\lambda _{{\ 3} }+5\, b\omega _{{1}}-5\, c\lambda _{{1}}). \end{eqnarray}$ $

(8.4)

Based on the SIVP and using derivative $ J $ respect to $ A $ and $ B $, one get

$\begin{eqnarray} \frac{\partial J}{\partial A} & = &\frac{1}{15}{k}^{2}\delta \, \mu \, -21\, \alpha \, b{k}^{2}{\mu }^{2}-21\, a{k}^{2}{\mu }^{2}-8\, \alpha \, b\delta \, k\mu -8\, a\delta \, k\mu +5\, ab\omega _{{2}}-5\, ac\lambda _{{2}} \\ &&+5\, {b}^{2}\omega _{{3}}-5\, bc\lambda _{{3}}+5\, b\omega _{{1}}-5\, c\lambda _{{1}})+\frac{1}{30}{k}^{2}{\delta }^{2}\mu \, \left( -8\, \alpha \, bk\mu -8\, ak\mu \right) = 0, \end{eqnarray}$ $

(8.5)

and

$\begin{eqnarray} \frac{\partial J}{\partial B} & = &\frac{1}{30}{k}^{2}{\delta }^{2}-21\, \alpha \, b{k}^{2}{\mu }^{2}-21\, a{\ k}^{2}{\mu }^{2}-8\, \alpha \, b\delta \, k\mu -8\, a\delta \, k\mu +5\, ab\omega _{{2}}-5\, ac\lambda _{{2}}+5\, {b}^{2}\omega _{{3}} \\ &&-5\, bc\lambda _{{3}}+5\, b\omega _{{1}}-5\, c\lambda _{{1}})+\frac{1}{30}{k} ^{2}{\delta }^{2}\mu \, \left( -42\, \alpha \, b{k}^{2}\mu -42\, a{k}^{2}\mu -8\, \alpha \, b\delta \, k-8\, a\delta \, k\right) = 0. \end{eqnarray}$ $

(8.6)

Solve the Eqs (8.5) and (8.6), become

$\begin{equation} \delta = \pm \frac{21}{2} \ \ {\frac{ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}} \ \ {\sqrt{ -\left( 21\, \alpha \, b+21\, a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{ b}^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) }} }, \ \ \ \end{equation}$ $

(8.7)

$ \mu = \pm \frac{1}{21} \ \ {\frac{\sqrt{-\left( 21\, \alpha \, b+21\, a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{b}^{2}\omega _{{3}}-bc\lambda _{{3} }+b\omega _{{1}}-c\lambda _{{1}}\right) }}{\left( \alpha \, b+a\right) k}}. $

The condition can be obtained as below

$\begin{equation} \left( \alpha \, b+a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) < 0. \end{equation}$ $

(8.8)

Finally, the solitary solution received by utilizing of SIVP can be reached as

$\begin{equation} \Psi (x, y, z, t) = \delta \ \mathrm{sech}\left[ k\mu (x+ay+bz-ct)\right] . \end{equation}$ $

(8.9)

8.2. Case II

We utilize the bright wave function as the following

$\begin{equation} u(\xi ) = \delta \ \mathrm{sech}^{2}(\mu \xi ). \end{equation}$ $

(8.10)

Hence, the stationary integral transforms to

$\begin{eqnarray} J & = &- \ \ {\frac{2\, {\delta }^{2}\mu (120\, \alpha \, b{k}^{2}{\mu }^{2}+120\, a{k} ^{2}{\mu }^{2}+35\, \alpha \, b\delta \, k\mu +35\, a\delta \, k\mu -14\, ab\omega _{{2}}} \ \ {105}} \\ &&+\frac{14\, ac\lambda _{{2}}-14\, {b}^{2}\omega _{{3}}+14\, bc\lambda _{{3} }-14\, b\omega _{{1}}+14\, c\lambda _{{1}})}{105}. \end{eqnarray}$ $

(8.11)

Based on the SIVP and using derivative $ J $ respect to $ A $ and $ B $, one get

$\begin{eqnarray} \frac{\partial J}{\partial A} & = &- \ \ {\frac{4\, \delta \, \mu (120\, \alpha \, b{k} ^{2}{\mu }^{2}+120\, a{k}^{2}{\mu }^{2}+35\, \alpha \, b\delta \, k\mu +35\, a\delta \, k\mu -14\, ab\omega _{{2}}+14\, ac\lambda _{{2}}} \ \ {105}} \\ &&+\frac{-14\, {b}^{2}\omega _{{3}}+14\, bc\lambda _{{3}}-14\, b\omega _{{1} }+14\, c\lambda _{{1}})-2\, {\delta }^{2}\mu \, \left( 35\, \alpha \, bk\mu +35\, ak\mu \right) }{105} = 0. \end{eqnarray}$ $

(8.12)

and

$\begin{eqnarray} \frac{\partial J}{\partial B} & = &- \ \ {\frac{2\, {\delta }^{2}(120\, \alpha \, b{k} ^{2}{\mu }^{2}+120\, a{\ k}^{2}{\mu }^{2}+35\, \alpha \, b\delta \, k\mu +35\, a\delta \, k\mu -14\, ab\omega _{{2}}+14\, ac\lambda _{{2}}-14\, {b} ^{2}\omega _{{3}}} \ \ {105}} \\ &&+\frac{+14\, bc\lambda _{{3}}-14\, b\omega _{{1}}+14\, c\lambda _{{1}})-2\, { \delta }^{2}\mu \, \left( 240\, \alpha \, b{k}^{2}\mu +240\, a{k}^{2}\mu +35\, \alpha \, b\delta \, k+35\, a\delta \, k\right) }{105} = 0. \end{eqnarray}$ $

(8.13)

Solve the Eqs (8.12) and (8.13), become

$\begin{equation} \delta = \pm \frac{48}{5} \ \ {\frac{ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}} \ \ {\sqrt{ -\left( 21\, \alpha \, b+21\, a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{ b}^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) }} }, \ \ \ \end{equation}$ $

(8.14)

$ \mu = \pm \frac{1}{30} \ \ {\frac{\sqrt{-\left( 21\, \alpha \, b+21\, a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{b}^{2}\omega _{{3}}-bc\lambda _{{3} }+b\omega _{{1}}-c\lambda _{{1}}\right) }}{\left( \alpha \, b+a\right) k}}. $

The condition of definition of the above relations can be expressed as

$\begin{equation} \left( \alpha \, b+a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) < 0. \end{equation}$ $

(8.15)

Finally, the solitary solution gained by utilizing of SIVP will be as

$\begin{equation} \Psi (x, y, z, t) = \delta \ \mathrm{sech}^{2}\left[ k\mu (x+ay+bz-ct)\right] . \end{equation}$ $

(8.16)

8.3. Case III

Assume the dark soliton wave solution be as the below case as

$\begin{equation} u(\xi ) = \delta \ \tanh ^{2}(\mu \xi ). \end{equation}$ $

(8.17)

Hence, the stationary integral transforms to

$\begin{eqnarray} J & = & \ \ {\frac{2\, {\delta }^{2}\mu (-120\, \alpha \, b{k}^{2}{\mu }^{2}-120\, a{k} ^{2}{\mu }^{2}+35\, \alpha \, b\delta \, k\mu +35\, a\delta \, k\mu +14\, ab\omega _{{2}}} \ \ {105}} \\ &&+\frac{-14\, ac\lambda _{{2}}+14\, {b}^{2}\omega _{{3}}-14\, bc\lambda _{{3} }+14\, b\omega _{{1}}-14\, c\lambda _{{1}})}{105}. \end{eqnarray}$ $

(8.18)

Based on the SIVP and using derivative $ J $ respect to $ A $ and $ B $, one get

$\begin{eqnarray} \frac{\partial J}{\partial \delta } & = & \ \ {\frac{4\, \delta \, \mu (-120\, \alpha \, b{k}^{2}{\mu }^{2}-120\, a{k}^{2}{\mu }^{2}+35\, \alpha \, b\delta \, k\mu +35\, a\delta \, k\mu +14\, ab\omega _{{2}}-14\, ac\lambda _{{2}}} \ \ {105}} \\ &&+\frac{+14\, {b}^{2}\omega _{{3}}-14\, bc\lambda _{{3}}+14\, b\omega _{{1} }-14\, c\lambda _{{1}})+2\, {\delta }^{2}\mu \, \left( 35\, \alpha \, bk\mu +35\, ak\mu \right) }{105} = 0, \end{eqnarray}$ $

(8.19)

and

$\begin{eqnarray} \frac{\partial J}{\partial \mu } & = & \ \ {\frac{2\, {\delta }^{2}(-120\, \alpha \, b{k }^{2}{\mu }^{2}-120\, a{\ k}^{2}{\mu }^{2}+35\, \alpha \, b\delta \, k\mu +35\, a\delta \, k\mu +14\, ab\omega _{{2}}-14\, ac\lambda _{{2}}+}{105}} \\ &&+\frac{14\, {b}^{2}\omega _{{3}}-14\, bc\lambda _{{3}}+14\, b\omega _{{1} }-14\, c\lambda _{{1}})+2\, {\delta }^{2}\mu \, \left( -240\, \alpha \, b{k} ^{2}\mu -240\, a{k}^{2}\mu +35\, \alpha \, b\delta \, k+35\, a\delta \, k\right) }{ 105} = 0. \end{eqnarray}$ $

(8.20)

Solve the Eqs (8.19) and (8.20), one get

$\begin{equation} \delta = \mp \frac{48}{5} \ \ {\frac{ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}} \ \ {\sqrt{ -\left( 21\, \alpha \, b+21\, a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{ b}^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) }} }, \ \ \ \end{equation}$ $

(8.21)

The condition of definition of the above relations can be presented the following form

$\begin{equation} \left( \alpha \, b+a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) < 0. \end{equation}$ $

(8.22)

Finally, the dark solution acquired by utilizing of SIVP will be as

$\begin{equation} \Psi (x, y, z, t) = \delta \ \mathrm{tanh}^{2}\left[ k\mu (x+ay+bz-ct)\right] . \end{equation}$ $

(8.23)

8.4. Case IV

Let the singular soliton wave solution be as the below case as

$\begin{equation} u(\xi ) = \delta \ csch^{2}(\mu \xi ). \end{equation}$ $

(8.24)

Then, the stationary integral transforms to

$\begin{eqnarray} J & = &- \ \ {\frac{2\, {\delta }^{2}\mu (-120\, \alpha \, b{k}^{2}{\mu }^{2}-120\, a{k }^{2}{\mu }^{2}-70\, \alpha \, b\delta \, k\mu -70\, a\delta \, k\mu +14\, ab\omega _{{2}}} \ \ {105}} \\ &&+\frac{-14\, ac\lambda _{{2}}+14\, {b}^{2}\omega _{{3}}-14\, bc\lambda _{{3} }+14\, b\omega _{{1}}-14\, c\lambda _{{1}})}{105}. \end{eqnarray}$ $

(8.25)

Based on the SIVP and using derivative $ J $ respect to $ A $ and $ B $, become

$\begin{eqnarray} \frac{\partial J}{\partial \delta } & = &- \ \ {\frac{4\, \delta \, \mu (-120\, \alpha \, b{k}^{2}{\mu }^{2}-120\, a{k}^{2}{\mu }^{2}-70\, \alpha \, b\delta \, k\mu -70\, a\delta \, k\mu +14\, ab\omega _{{2}}-14\, ac\lambda _{{2} }}{105}} \\ &&+\frac{+14\, {b}^{2}\omega _{{3}}-14\, bc\lambda _{{3}}+14\, b\omega _{{1} }-14\, c\lambda _{{1}})-2\, {\delta }^{2}\mu \, \left( -70\, \alpha \, bk\mu -70\, ak\mu \right) }{105} = 0 \end{eqnarray}$ $

(8.26)

and

$\begin{eqnarray} \frac{\partial J}{\partial \mu } & = &- \ \ {\frac{2\, {\delta }^{2}(-120\, \alpha \, b{ k}^{2}{\mu }^{2}-120\, a{k}^{2}{\mu }^{2}-70\, \alpha \, b\delta \, k\mu -70\, a\delta \, k\mu +14\, ab\omega _{{2}}-14\, ac\lambda _{{2}}+14\, {b} ^{2}\omega _{{3}}} \ \ {105}} \\ &&+\frac{-14\, bc\lambda _{{3}}+14\, b\omega _{{1}}-14\, c\lambda _{{1}})-2\, { \delta }^{2}\mu \, \left( -240\, \alpha \, b{k}^{2}\mu -240\, a{k}^{2}\mu -70\, \alpha \, b\delta \, k-70\, a\delta \, k\right) }{105} = 0. \end{eqnarray}$ $

(8.27)

Solve the Eqs (8.26) and (8.27), one get

$\begin{equation} \delta = \pm \frac{24}{5} \ \ {\frac{ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}} \ \ {\sqrt{ -\left( 21\, \alpha \, b+21\, a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{ b}^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) }} }, \ \ \ \end{equation}$ $

(8.28)

The condition of definition of the above relations can be presented as the below form as

$\begin{equation} \left( \alpha \, b+a\right) \left( ab\omega _{{2}}-ac\lambda _{{2}}+{b} ^{2}\omega _{{3}}-bc\lambda _{{3}}+b\omega _{{1}}-c\lambda _{{1}}\right) < 0. \end{equation}$ $

(8.29)

Finally, the singular solution acquired by utilizing of SIVP will be as

$\begin{equation} \Psi (x, y, z, t) = \delta \ \mathrm{csch}^{2}\left[ k\mu (x+ay+bz-ct)\right] . \end{equation}$ $

(8.30)

9. Conclusion

In this article, the MEFM employed for searching the MSSs for the gKP equation, which contains 1-wave, 2-wave, and 3-wave solutions. The periodic wave, cross-kink, and solitary wave solutions have been obtained. In continuing, the modulation instability applied to discuss the stability of earned solutions. It is quite visible that these novel schemes have plenty of family solutions containing rational exponential, hyperbolic, and periodic functions with selecting particular parameters. Also, the semi-inverse variational principle will be used for the gKP equation. Four major cases containing the solitary, bright, dark and singular wave solutions were studied from four different ansatzes.

By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained. Via various curve plots, density plot and three-dimensional plots, dynamical characteristics of these rouge waves are exhibited. Because of the strong nonlinear characteristic of Hirota bilinear method, the test function constructed by the Hirota operator, which can be regarded as the test function constructed by considered model. The results are beneficial to the study of the plasma, optics, acoustics, fluid dynamics and fluid mechanics. All computations in this paper have been employed quickly with the help of the Maple 18. Moreover, the method applied in this paper provides an effective tool to obtain exact solutions of nonlinear system and can in common use for other NLEEs.

Acknowledgments

This work is supported by the Education and scientific research project for young and middle-aged teachers of Fujian Province (No. JAT.190666-No.JAT200469)

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1]	Cowan N (1995) Attention and memory: An integrated framework. Oxford, England: Oxford University Press.
[2]	Jonides J, Lacey SC, Nee DE (2005) Processes of working memory in mind and brain. Curr Dir Psychol 14: 2-5. doi: 10.1111/j.0963-7214.2005.00323.x
[3]	Lewis-Peacock JA, Postle BR (2008) Temporary activation of long-term memory supports working memory. J Neurosci 28: 8765-8771. doi: 10.1523/JNEUROSCI.1953-08.2008
[4]	Ruchkin DS, Grafman J, Cameron K, et al. (2003) Working memory retention systems: A state of activated long-term memory. Behav Brain Sci 26: 709-728.
[5]	Zimmer HD (2008) Visual and spatial working memory: from boxes to networks. Neurosci Biobehav Rev 32: 1373-1395. doi: 10.1016/j.neubiorev.2008.05.016
[6]	Cowan N (1984) On short and long auditory stores. Psychol Bull 96: 341-370. doi: 10.1037/0033-2909.96.2.341
[7]	Massaro DW (1975) Backward recognition masking. J Acoust Soc Am 58: 1059-1065. doi: 10.1121/1.380765
[8]	Sperling G (1960) The information available in brief visual presentations. Psychol Monogr 74: 1-29.
[9]	Massaro DW (1972) Preperceptual images, processing time, and perceptual units in auditory perception. Psychol Rev 79: 124-145. doi: 10.1037/h0032264
[10]	Turvey MT (1973) On peripheral and central processes in vision: inferences from an information processing analysis of masking with patterned stimuli. Psychol Rev 80: 1-52. doi: 10.1037/h0033872
[11]	Vogel EK, Woodman GF, Luck SJ (2006) The time course of consolidation in visual working memory. J Exp Psychol-Hum Percept Perform 32: 1436-1451. doi: 10.1037/0096-1523.32.6.1436
[12]	Woodman GF, Vogel EK (2005) Fractionating working memory consolidation and maintenance are independent processes. Psychol Sci 16: 106-113. doi: 10.1111/j.0956-7976.2005.00790.x
[13]	Woodman GF, Vogel EK (2008) Selective storage and maintenance of an object's features in visual working memory. Psychon Bull Rev 15: 223-229. doi: 10.3758/PBR.15.1.223
[14]	Bradshaw GL, Anderson JR (1982) Elaborative encoding as an explanation of levels of processing. J Verb Learn Verb Beh 21: 165-174. 15. Craik FIM, Lockhart RS (1972) Levels of processing: A framework for memory research. J Verb Learn Verb Beh 11: 671-684. doi: 10.1016/S0022-5371(72)80001-X
[15]	16. Craik FI, Tulving E (1975) Depth of processing and the retention of words in episodic memory. J Exp Psychol Gen 104: 268-294.
[16]	17. Nieuwenstein M, Wyble B (2014) Beyond a mask and against the bottleneck: Retroactive dual-task interference during working memory consolidation of a masked visual target. J Exp Psychol Gen 143: 1409-1427.
[17]	18. Ricker TJ, Cowan N (2014) Differences in presentation methods in working memory procedures: A matter of working memory consolidation. J Exp Psychol Learn Mem Cogn 40: 4428.
[18]	19. Jolicoeur P, Dell'Acqua R (1998) The demonstration of short-term consolidation. Cogn Psychol 36: 138-202. doi: 10.1006/cogp.1998.0684
[19]	20. Baddeley AD (6) Working memory. New York, NY: Oxford University Press.
[20]	21. Barrouillet P, Camos V (2) As time goes by: Temporal constraints in working memory. Curr Dir Psychol 21: 413-419. doi: 10.1177/0963721412459513
[21]	22. Davelaar EJ, Goshen-Gottstein Y, Ashkenazi A, et al. (2005) The demise of short-term memory revisited: empirical and computational investigations of recency effects. Psychol Rev 112: 3-42. doi: 10.1037/0033-295X.112.1.3
[22]	23. Unsworth N, Engle RW (2007) The nature of individual differences in working memory capacity: Active maintenance in primary memory and controlled search from secondary memory. Psychol Rev 114: 104-132. doi: 10.1037/0033-295X.114.1.104
[23]	24. Brown GDA, Neath I, Chater N (2007) A temporal ratio model of memory. Psychol Rev 114: 539-576. doi: 10.1037/0033-295X.114.3.539
[24]	25. Farrell S (2012) Temporal clustering and sequencing in short-term memory and episodic memory. Psychol Rev 119: 223-271. doi: 10.1037/a0027371
[25]	26. Nairne JS (2002) Remembering over the short term: The case against the standard model. Annu Rev Psychol 53: 53-81. doi: 10.1146/annurev.psych.53.100901.135131
[26]	27. Tiitinen H, Reinikainen PMK, Näätänen R (1994) Attentive novelty detection in humans is governed by pre-attentive sensory memory. Nature 372: 90-92. doi: 10.1038/372090a0
[27]	28. Cowan N (1988) Evolving conceptions of memory storage, selective attention, and their mutual constraints within the human information processing system. Psychol Bull 104: 163-191.
[28]	29. Khader P, Burke M, Bien S, et al. (2005) Content-specific activation during associative long-term memory retrieval. Neuroimage 27: 805-816. doi: 10.1016/j.neuroimage.2005.05.006
[29]	30. Craik FI, Govoni R, Naveh-Benjamin M, et al. (1996) The effects of divided attention on encoding and retrieval processes in human memory. J Exp Psychol Gen 125: 159-180. doi: 10.1037/0096-3445.125.2.159
[30]	31. Naveh-Benjamin M, Craik FIM, Gavrilescu D, et al. (2000) Asymmetry between encoding and retrieval processes: Evidence from divided attention and a calibration analysis. Mem Cognit 28: 965-976.
[31]	32. Massaro DW (1975) Experimental psychology and information processing. Chicago, IL: Rand McNally.
[32]	33. Baddeley AD (2000) The episodic buffer: A new component of working memory? Trends Cogn Sci 4: 417-423. doi: 10.1016/S1364-6613(00)01538-2
[33]	34. Logie RH (2009) Working memory, In: Bayne T, Cleeremans T, Wilken P, The Oxford companion to consciousness, Oxford, UK: Oxford University Press, 667-670.
[34]	35. Massaro DW (1970) Perceptual processes and forgetting in memory tasks. Psychol Rev 77: 557-567.
[35]	36. Saults JS, Cowan N (2007) A central capacity limit to the simultaneous storage of visual and auditory arrays in working memory. J Exp Psychol Gen 136: 663-684.
[36]	37. Sligte IG, Scholte HS, Lamme VA (2008) Are there multiple visual short-term memory stores? PLoS One 3: e1699. doi: 10.1371/journal.pone.0001699
[37]	38. Pinto Y, Sligte IG, Shapiro KL, et al. (2013) Fragile visual short-term memory is an object-based and location-specific store. Psychon Bull Rev 20: 732-739. doi: 10.3758/s13423-013-0393-4
[38]	39. Cowan N, Ricker TJ, Clark KM, et al. (2015) Knowledge cannot explain the developmental growth of working memory capacity. Dev Sci 18: 132-145. doi: 10.1111/desc.12197
[39]	40. McKeown D, Mercer T (2012) Short-term forgetting without interference. J Exp Psychol Learn Mem Cogn 38: 1057-1068.
[40]	41. Morey CC, Bieler M (2013) Visual short-term memory always requires general attention. Psychon Bull Rev 20: 163-170. doi: 10.3758/s13423-012-0313-z
[41]	42. Ricker TJ, Cowan N (2010) Loss of visual working memory within seconds: The combined use of refreshable and non-refreshable features. J Exp Psychol Learn Mem Cogn 36: 1355-1368.
[42]	43. Ricker TJ, Spiegel LR, Cowan N (2014) Time-based loss in visual short-term memory is from trace decay, not temporal distinctiveness. J Exp Psychol Learn Mem Cogn 40: 1510-1523. doi: 10.1037/xlm0000018
[43]	44. Ricker TJ, Vergauwe E, Hinrichs GA, et al. (2015) No recovery of memory when cognitive load is decreased. J Exp Psychol Learn Mem Cogn 41: 872-880. doi: 10.1037/xlm0000084
[44]	45. Vergauwe E, Camos V, Barrouillet P (2014) The impact of storage on processing: How is information maintained in working memory? J Exp Psychol Learn Mem Cogn 40: 1072-1095. doi: 10.1037/a0035779
[45]	46. Woodman GF, Vogel EK, Luck SJ (2012) Flexibility in visual working memory: Accurate change detection in the face of irrelevant variations in position. Vis Cogn 20: 1-28. doi: 10.1080/13506285.2011.630694
[46]	47. Zhang W, Luck SJ (2009) Sudden death and gradual decay in visual working memory. Psychol Sci 20: 423-428. doi: 10.1111/j.1467-9280.2009.02322.x
[47]	48. Lewandowsky S, Geiger SM, Oberauer K (2008) Interference-based forgetting in verbal short-term memory. J Mem Lang 59: 200-222. doi: 10.1016/j.jml.2008.04.004
[48]	49. Oberauer K, Lewandowsky S, Farrell S, et al. (2012) Modeling working memory: An interference model of complex span. Psychon Bull Rev 19: 779-819. doi: 10.3758/s13423-012-0272-4
[49]	50. Barrouillet P, Bernardin S, Camos V (2004) Time constraints and resource sharing in adults' working memory spans. J Exp Psychol Gen 133: 83-100. doi: 10.1037/0096-3445.133.1.83
[50]	51. Vergauwe E, Barrouillet P, Camos V (2010) Do mental processes share a domain-general resource? Psychol Sci 21: 384-390. doi: 10.1177/0956797610361340
[51]	52. Oberauer K, Lewandowsky S (2011) Modeling working memory: a computational implementation of the Time-Based Resource-Sharing theory. Psychon Bull Rev 18: 10-45. doi: 10.3758/s13423-010-0020-6
[52]	53. Oberauer K, Lewandowsky S (2013) Evidence against decay in verbal working memory. J Exp Psychol Gen 142: 380-411. doi: 10.1037/a0029588
[53]	54. Crowder RG (1976) Principles of learning and memory. Hillsdale, NJ: Erlbaum.
[54]	55. Naveh-Benjamin M, Jonides J (1984) Maintenance rehearsal: A two component analysis. J Exp Psychol Learn Mem Cogn 10: 369-385.
[55]	56. Johnson MK, Reeder JA, Raye CL, et al. (2002) Second thoughts versus second looks: An age-related deficit in reflectively refreshing just-activated information. Psychol Sci 13: 64-67. doi: 10.1111/1467-9280.00411
[56]	57. Raye CL, Johnson MK, Mitchell KJ, Greene EJ, Johnson MR (2007) Refreshing: A minimal executive function. Cortex 43: 135-145. doi: 10.1016/S0010-9452(08)70451-9
[57]	58. Portrat S, Lemaire B (2015) Is attentional refreshing in working memory sequential? A computational modeling approach. Cognitive Computation 7: 333-345.
[58]	59. Vergauwe E, Cowan N (2014) A common short-term memory retrieval rate may describe many cognitive procedures. Front Hum Neurosci 8.
[59]	60. Bayliss DM, Bogdanovs J, Jarrold C (2015) Consolidating working memory: Distinguishing the effects of consolidation, rehearsal and attentional refreshing in a working memory span task. J Mem Lang 81: 34-50. doi: 10.1016/j.jml.2014.12.004
[60]	61. Stevanovski B, Jolicoeur P (2007) Visual short-term memory: Central capacity limitations in short-term consolidation. Vis Cogn 15: 532-563. doi: 10.1080/13506280600871917
[61]	62. Eichenbaum H (2000) A cortical–hippocampal system for declarative memory. Nat Rev Neurosci 1: 41-50. doi: 10.1038/35036213
[62]	63. Remondes M, Schuman EM (2004) Role for a cortical input to hippocampal area CA1 in the consolidation of a long-term memory. Nature 431: 699-703. doi: 10.1038/nature02965
[63]	64. Squire LR, Alvarez P (1995) Retrograde amnesia and memory consolidation: A neurobiological perspective. Curr Opin Neurobiol 5: 169-177.
[64]	65. Bliss TV, Collingridge GL (1993) A synaptic model of memory: Long-term potentiation in the hippocampus. Nature 361: 31-39. doi: 10.1038/361031a0
[65]	66. Teyler TJ, DiScenna P (1987) Long-term potentiation. Annu Rev Neurosci 10: 131-161. doi: 10.1146/annurev.ne.10.030187.001023
[66]	67. Hampson RE, Deadwyler SA (2000) Cannabinoids reveal the necessity of hippocampal neural encoding for short-term memory in rats. J Neurosci 20: 8932-8942.
[67]	68. Mitchell KJ, Johnson MK, Raye CL, et al. (2000) fMRI evidence of age-related hippocampal dysfunction in feature binding in working memory. Cogn Brain Res 10: 197-206. doi: 10.1016/S0926-6410(00)00029-X
[68]	69. Nichols EA, KaoYC, Verfaellie M, et al. (2006) Working memory and long‐term memory for faces: Evidence from fMRI and global amnesia for involvement of the medial temporal lobes. Hippocampus 16: 604-616. doi: 10.1002/hipo.20190
[69]	70. Vertes RP (2005) Hippocampal theta rhythm: A tag for short‐term memory. Hippocampus 15: 923-935. doi: 10.1002/hipo.20118
[70]	71. Cowan N, Li D, Moffitt A, Becker TM, et al. (2011) A neural region of abstract working memory. J Cogn Neurosci 23: 2852-2863. doi: 10.1162/jocn.2011.21625
[71]	72. Emrich SM, Riggall AC, LaRocque JJ, et al. (2013) Distributed patterns of activity in sensory cortex reflect the precision of multiple items maintained in visual short-term memory. J Neurosci 33: 6516-6523. doi: 10.1523/JNEUROSCI.5732-12.2013
[72]	73. Xu Y, Chun MM (2006) Dissociable neural mechanisms supporting visual short-term memory for objects. Nature 440: 91-95.
[73]	74. Todd JJ, Marois R (2004) Capacity limit of visual short-term memory in human posterior parietal cortex. Nature 428: 751-754.
[74]	75. Crick F, Koch C (1990) Toward a neurobiological theory of consciousness. Semin Neurosci 2: 263-275.
[75]	76. Engel AK, Singer W (2001) Temporal binding and the neural correlates of sensory awareness. Trends Cogn Sci 5: 16-25. doi: 10.1016/S1364-6613(00)01568-0
[76]	77. Lisman JE, Idiart MA (1995) Storage of 7+/-2 short-term memories in oscillatory subcycles. Science 267: 1512-1515.
[77]	78. Lisman JE, Jensen O (2013) The theta-gamma neural code. Neuron 1002-1016. doi: 10.1016/j.neuron.2013.03.007
[78]	79. Shipstead Z, Lindsey DR, Marshall RL, et al. (2014) The mechanisms of working memory capacity: Primary memory, secondary memory, and attention control. J Mem Lang 72: 116-141. doi: 10.1016/j.jml.2014.01.004

This article has been cited by:

1.	Chaudry Masood Khalique, Oke Davies Adeyemo, Kentse Maefo, Symmetry solutions and conservation laws of a new generalized 2D Bogoyavlensky-Konopelchenko equation of plasma physics, 2022, 7, 2473-6988, 9767, 10.3934/math.2022544
2.	K. Hosseini, D. Baleanu, S. Rezapour, S. Salahshour, M. Mirzazadeh, M. Samavat, Multi-complexiton and positive multi-complexiton structures to a generalized B-type Kadomtsev−Petviashvili equation, 2022, 24680133, 10.1016/j.joes.2022.06.020
3.	Abdul-Majid Wazwaz, Naisa S. Alatawi, Wedad Albalawi, S. A. El-Tantawy, Painlevé analysis for a new (3 +1 )-dimensional KP equation: Multiple-soliton and lump solutions, 2022, 140, 0295-5075, 52002, 10.1209/0295-5075/aca49f
4.	Yan Zhu, Chuyu Huang, Junjie Li, Runfa Zhang, Lump solitions, fractal soliton solutions, superposed periodic wave solutions and bright-dark soliton solutions of the generalized (3+1)-dimensional KP equation via BNNM, 2024, 112, 0924-090X, 17345, 10.1007/s11071-024-09911-2
5.	Adisie Fenta Agmas, Fasika Wondimu Gelu, Meselech Chima Fino, A robust, exponentially fitted higher-order numerical method for a two-parameter singularly perturbed boundary value problem, 2025, 10, 2297-4687, 10.3389/fams.2024.1501271

Reader Comments

Your name:*

Email:*
© 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Neuroscience

3.1 4.2

Metrics

Article views(9783) PDF downloads(1591) Cited by(24)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Neuroscience

The Role of Short-term Consolidation in Memory Persistence

Related Papers:

Abstract

1. Introduction

2. Multiple Exp-function method

3. MSSs for the (3+1) gKP equation

3.1. Option I: 1-wave solution

3.2. Option II: 2-wave solutions

3.3. Option III: Triple-wave solutions

4. Novel periodic wave solutions

5. Novel cross-kink wave solutions

6. Novel solitary wave solutions

7. Stability analysis of Eq (1.8)

8. Usage of SIVP for Eq (1.8)

8.1. Case I

8.2. Case II

8.3. Case III

8.4. Case IV

9. Conclusion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Multiple Exp-function method

3. MSSs for the (3+1) gKP equation

3.1. Option I: 1-wave solution

3.2. Option II: 2-wave solutions

3.3. Option III: Triple-wave solutions

4. Novel periodic wave solutions

5. Novel cross-kink wave solutions

6. Novel solitary wave solutions

7. Stability analysis of Eq (1.8)

8. Usage of SIVP for Eq (1.8)

8.1. Case I

8.2. Case II

8.3. Case III

8.4. Case IV

9. Conclusion

Acknowledgments

Conflict of interest

References