Output-based event-triggered control for discrete-time systems with three types of performance analysis

Hyung Tae Choi; Hae Yeon Park; Jung Hoon Kim; Hyung Tae Choi; Hae Yeon Park; Jung Hoon Kim

doi:10.3934/math.2023873

AIMS Mathematics

2023, Volume 8, Issue 7: 17091-17111. doi: 10.3934/math.2023873

Previous Article Next Article

Research article Special Issues

Output-based event-triggered control for discrete-time systems with three types of performance analysis

1.
Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
2.
Institute for Convergence Research and Education in Advanced Technology, Yonsei University, Incheon 21983, Republic of Korea

Received: 15 March 2023 Revised: 25 April 2023 Accepted: 09 May 2023 Published: 17 May 2023
MSC : 39A06, 93C55, 93D20, 93D25, 93D30

This paper considers an output-based event-triggered control approach for discrete-time systems and proposes three new types of performance measures under unknown disturbances. These measures are motivated by the fact that signals in practical systems are often associated with bounded energy or bounded magnitude, and they should be described in the $\ell_{2}$ and $\ell_\infty$ spaces, respectively. More precisely, three performance measures from $\ell_{q}$ to $\ell_{p}$ , denoted by the $\ell_{p/q}$ performances with $(p, q) = (2, 2), \ (\infty, 2)$ and $(\infty, \infty)$ , are considered for event-triggered systems (ETSs) in which the corresponding event-trigger mechanism is defined as a function from the measured output of the plant to the input of the dynamic output-feedback controller with the triggering parameter $\sigma (>0)$ . Such a selection of the pair $(p, q)$ represents the $\ell_{p/q}$ performances to be bounded and well-defined, and the three measures are natural extensions of those in the conventional feedback control, such as the $H_\infty$ , generalized $H_2$ and $\ell_1$ norms. We first derive the corresponding closed-form representation with respect to the relevant ETSs in terms of a piecewise linear difference equation. The asymptotic stability condition for the ETSs is then derived through the linear matrix inequality approach by developing an adequate piecewise quadratic Lyapunov function. This stability criterion is further extended to compute the $\ell_{p/q}$ performances. Finally, a numerical example is given to verify the effectiveness of the overall arguments in both the theoretical and practical aspects, especially for the trade-off relation between the communication costs and $\ell_{p/q}$ performances.

Keywords:

Citation: Hyung Tae Choi, Hae Yeon Park, Jung Hoon Kim. Output-based event-triggered control for discrete-time systems with three types of performance analysis[J]. AIMS Mathematics, 2023, 8(7): 17091-17111. doi: 10.3934/math.2023873

Related Papers:

[1]	Jun Wang, Dan Wang, Yuan Yuan . Research on low-carbon closed-loop supply chain strategy based on differential games-dynamic optimization analysis of new and remanufactured products. AIMS Mathematics, 2024, 9(11): 32076-32101. doi: 10.3934/math.20241540
[2]	Zifu Fan, Kexin Fan, Wei Zhang . Research on the data integration strategy of local governments and enterprises under central government subsidies. AIMS Mathematics, 2022, 7(6): 10143-10164. doi: 10.3934/math.2022564
[3]	Chengli Hu, Ping Liu, Hongtao Yang, Shi Yin, Kifayat Ullah . A novel evolution model to investigate the collaborative innovation mechanism of green intelligent building materials enterprises for construction 5.0. AIMS Mathematics, 2023, 8(4): 8117-8143. doi: 10.3934/math.2023410
[4]	Zifu Fan, Youpeng Tao, Wei Zhang, Kexin Fan, Jiaojiao Cheng . Research on open and shared data from government-enterprise cooperation based on a stochastic differential game. AIMS Mathematics, 2023, 8(2): 4726-4752. doi: 10.3934/math.2023234
[5]	Shuai Huang, Youwu Lin, Jing Zhang, Pei Wang . Chance-constrained approach for decentralized supply chain network under uncertain cost. AIMS Mathematics, 2023, 8(5): 12217-12238. doi: 10.3934/math.2023616
[6]	Jewaidu Rilwan, Poom Kumam, Idris Ahmed . Overtaking optimality in a discrete-time advertising game. AIMS Mathematics, 2022, 7(1): 552-568. doi: 10.3934/math.2022035
[7]	Jamilu Adamu, Kanikar Muangchoo, Abbas Ja'afaru Badakaya, Jewaidu Rilwan . On pursuit-evasion differential game problem in a Hilbert space. AIMS Mathematics, 2020, 5(6): 7467-7479. doi: 10.3934/math.2020478
[8]	Hamidou Tembine . Mean-field-type games. AIMS Mathematics, 2017, 2(4): 706-735. doi: 10.3934/Math.2017.4.706
[9]	Lin Xu, Linlin Wang, Hao Wang, Liming Zhang . Optimal investment game for two regulated players with regime switching. AIMS Mathematics, 2024, 9(12): 34674-34704. doi: 10.3934/math.20241651
[10]	Bei Yuan . Mathematical modeling of intellectual capital asymmetric information game in financial enterprises. AIMS Mathematics, 2024, 9(3): 5708-5721. doi: 10.3934/math.2024277

Abstract

1. Introduction

The soliton solutions of nonlinear evolution equations (NLEE) play an important role to understand the various advantages of physical phenomena in fluids, plasmas, etc ^[1,2]. Rational solutions are a particular type of soliton solutions, which include lump and rogue waves.

A rogue wave which is a type of rational solution is isolated. It is reported in ^[3,4] that the mean height of rogue waves is at least double times the height of the neighboring waves. Rogue waves come from nowhere and disappear with no trace ^[5,6] and superfluids ^[7]. The applications of rogue waves with their rational solutions for the Boussinesq equation can be found in ^[8].

Rogue waves can be seen in the thick ocean ^[7,9], water tanks ^[10,11], and optical fibers ^[10,12].

The purpose for the rogue wave is to understand the physics of the huge waves appearance and its relations to the environmental conditions (wind and atmospheric pressure).

In the literature, many methods were proposed to derive the rogue wave solutions such as an inverse scattering method ^[13], Hirota bilinear method ^[14], Darboux transformation method ^[15], B $\ddot{a}$ cklund transformation method ^[16], variational direct method, simplified extended tanh-function method, Exp-function method, extended rational Sin-Cos and sinh-cosh methods. In ^[17] the author derives the periodic wave solution for the fractional complex nonlinear Fokas-Lenells equation via an ancient Chinese algorithm so called the Ying Bu Zu Shu. Also in ^[18] the authors deduced the abundant solitons and periodic solutions of the (1+2)-dimensional chiral nonlinear Schr $\ddot{o}$ dinger equation using the extended He's variational method. In ^[19] the same authors gave the explicit solutions for the Benney-Luke equation in dark solitary, dark-like solitary, kinky dark solitary and periodic wave solutions by the variational direct method (VDM).

The KP equation is a nonlinear partial differential equation in one temporal and two or three spatial coordinate, describes the evolution of nonlinear long waves for small amplitudes.

Furthermore, the (KP) equation was proposed to deal with slowly varying perturbation wave in dispersion media. This equation has been studied in a variety of scientific fields, such as solid state, physics, plasma physics, fiber optics, propagation of waves, oceanography ^[20].

The importance of the nonlinear terms is to obtain the rogue waves which are nonlinear waves phenomena, so they can be represented with a variety of nonlinear wave equations. Bilinear equation was obtained for soliton equation only and can be used to handle the nonlinear wave equations.

The novelty of this work is handling the higher-order rogue wave solution by the (KP)equation in form of two (3+1)-dimensional extensions

The multi-rogue wave solutions are found for the (3+1)-dimensional Jimbo-Miwa equation ^[21] and (3+1)-dimensional Hirota bilinear equation based on the bilinear form for this equation using a symbolic computation approach ^[22]. It is noted that there are similarity between the results in the references ^[21,22] and the obtained results in this paper which prove the correctness of the proposed approach.

N-soliton solutions are obtained based on the simplified Hirota approach and kinky-lump breather, combo line kink and kinky-lump breather are derived for (3+1)-dimensional Sharma-Tasso-Olver-like (STOL) model ^[23]. By the bilinear form for the extended BKPA-Boussinesq equation the abundant breather waves, multi-shocks waves and localized excitation solutions are obtained ^[24].

The symbolic computation approach method was chosen because it is a simple method to obtain the higher order rogue wave solution without need to obtain a Darboux transformation ^[25].

Multi rogue waves of the Boussinesq type equation ^[26,27,28].

A rogue wave can be formed when wave energy is focused, usually during a storm. When the storm produces waves that go against the prevailing ocean current, the wave frequency shortens.

In ^[29] the authors constructed the multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation in fluid dynamics, in ^[30] the authors studied (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. By adding some new constraints to the N-soliton solutions and the resonance Y-type soliton and in ^[31] the authors investigated the multiple lump solutions for the generalized (3+1)-dimensional KP equation. With the aid of the variable transformation

In this work, section 2 introduces the description of the symbolic computation approach ^[32]. Section 3 introduces the multi-rogue waves and bilinear system of the first extended (3+1)-dimensional KP equation. The first, second, and third-order rogue waves for this equation are derived in subsections 3.1–3.3. Section 4 introduces the multi-rogue waves and bilinear form of the second extended (3+1)-dimensional KP equation. Subsections 4.1–4.3 introduces the derivations of the first, second, and third-order rogue waves for this equation. Section 5 introduces the results and conclusions of the outcome results.

2. Materials and methods

The nonlinear partial differential equations (NLEEs) are:

$\begin{eqnarray} &&H(u, u_{t}, u_{x}, u_{y}, u_{z}, u_{xt}, u_{xy}, u_{xz}...) = 0. \end{eqnarray}$

(2.1)

Here, $H$ is a function in $u(x, y, z, t)$ and its derivatives.

The main steps of the proposed approach are:

Step 1. Following the Painlève analysis

$\begin{eqnarray} &&u(x, y, z, t) = u(\zeta, z). \end{eqnarray}$

(2.2)

Step 2. By using (2.2) in NLEEs (2.1), the Hirota's bilinear form is:

$\begin{eqnarray} &&F(D_\zeta, D_z) = 0, \end{eqnarray}$

(2.3)

where $\zeta = x+y-et$ . The D-operator is defined by ^[33]

$\begin{eqnarray} && D_x^k D_y^m D_z^n D_t^l g(x, y, z, t)\cdot f(x, y, z, t) = (-1)^{k+m+n+l} (\frac{\partial}{\partial x'}-\frac{\partial}{\partial x})^k (\frac{\partial}{\partial y'}-\frac{\partial}{\partial y})^m (\frac{\partial}{\partial z'}-\frac{\partial}{\partial z})^n\\&& \qquad\qquad\qquad\qquad\qquad\qquad\quad(\frac{\partial}{\partial t'}-\frac{\partial}{\partial t})^l [f(x, y, z, t)g(x', y', z', t')]\mid_{x' = x, y' = y, z' = z, t' = t}. \end{eqnarray}$

(2.4)

Step 3. Let

$\begin{eqnarray} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!F = G_{n+1}(\zeta, z;\alpha, \beta) = 2\, \alpha\, zP_{{n}} \left( \zeta , z \right) +F_{{n+1}} \left( \zeta , z \right) +2\, \beta\, \left( \zeta \right) \, Q_{{n}} \left( \zeta , z \right) + \left( {\alpha}^{2}+{\beta}^{2} \right) F_{{n-1}} \left( \zeta , z \right), \end{eqnarray}$

(2.5)

where

$\begin{eqnarray} &&\!\!\!\!F_{n}(\zeta, z;\alpha, \beta) = \sum _{k = 0}^{\frac{1}{2}\, n \left( n+1 \right) } \left( \sum _{i = 0}^{k}{z}^{2\, i}a_{{n \left( n+1 \right) -2\, k, 2\, i}}{\zeta }^{n \left( n+1 \right) -2\, k} \right), \\&&P_n(\zeta, z) = \sum _{k = 0}^{\frac{1}{2}\, n \left( n+1 \right) } \left( \sum _{i = 0}^{k}{z}^{2\, i}b_{{n \left( n+1 \right) -2\, k, 2\, i}}{\zeta }^{n \left( n+1 \right) -2\, k} \right), \\&&Q_n(\zeta, z) = \sum _{k = 0}^{\frac{1}{2}\, n \left( n+1 \right) } \left( \sum _{i = 0}^{k}{z}^{2\, i}c_{{n \left( n+1 \right) -2\, k, 2\, i}}{\zeta }^{n \left( n+1 \right) -2\, k} \right). \end{eqnarray}$

(2.6)

$F_0 = 1, F_{-1} = P_0 = Q_0 = 0$ , $\alpha, \beta, a_{m, l}, b_{m, l}$ and $c_{m.l}, (m, l = 0, 2, 4, ...., n(n+1))$ are real numbers. $\alpha, \beta$ are used to control the rogue-wave center.

Step 4. By substituting (2.5) into (2.3) and taking the coefficients of $z$ and $\zeta$ equal to zero, a system of polynomials is obtained.

Step 5. For getting the multi rogue wave solutions in terms of $z$ and $\zeta$ , the values of $a_{m, l}, b_{m, l}$ and $c_{m, l}$ are substituted into (2.5).

3. The bilinear form and multi rogue waves for the first extended (3+1)-dimensional KP equation

The first extended (3+1)-dimensional KP equation ^[34]:

$\begin{eqnarray} &&(u_{t}+\delta uu_x+\mu u_{xxx})_x+\chi (u_{xx}+u_{yy}+u_{zz}) = 0, \end{eqnarray}$

(3.1)

where $\delta, \mu$ and $\chi$ are plasma parameters. And $u$ is a wave amplitude functions in $x, y, z$ and $t$ . The rogue-waves solutions for (3.1) can be obtained by finding the Hirota bilinear form.

In (3.1) setting $\zeta = x+d y-et$ ,

$\begin{eqnarray} &&\mu u_{\zeta\zeta\zeta\zeta}+({d}^{2}\chi+\delta\, u -e+\chi) u_{\zeta\zeta}+\delta u_\zeta^2+\chi u_{zz} = 0. \end{eqnarray}$

(3.2)

Using the following variable transformation

$\begin{eqnarray} &&u = u_0+\frac{12\mu}{\delta} (\ln F)_{\zeta\zeta}. \end{eqnarray}$

(3.3)

Where $u_0$ is a constant, the Hirota bilinear form for (3.1) is obtained by substituting (3.3) into (3.2):

$\begin{eqnarray} &&(\mu D_{\zeta}^4+({d}^{2}\chi+\delta\, u_0 -e+\chi)D_\zeta^2+\chi D_z^2)F \cdot F = 0. \end{eqnarray}$

(3.4)

If $\chi = 1, \mu = -(1/3), e = {d}^{2}+\delta\, u_0$ , then (3.1) converts to the Boussinesq equation ^[8].

The multi rogue wave solutions of the first extended (3+1)-dimensional (KP) equation (3.1) can be obtained as follows.

3.1. First-order rogue waves $n=0$

Here, $F$ is selected as

$\begin{eqnarray} &&F = G_1 = a_{{2, 0}}{\zeta }^{2}+a_{{0, 2}}{z}^{2}+a_{{0, 0}}. \end{eqnarray}$

(3.5)

Let $a_{2, 0} = 1$ with no loss of generalization.

The coefficients $a_{0, 0}$ and $a_{0, 2}$ are obtained by substituting (3.5) into (3.4) and taken zero value for the coefficients of all powers of $z$ and $\zeta$

$\begin{eqnarray} &&a_{{0, 0}} = -\, {\frac {3\mu}{{d}^{2}\chi+\delta\, u_{{0}}+\chi-e}} , \qquad \qquad a_{{0, 2}} = {\frac {{d}^{2}\chi+\delta\, u_{{0}}+\chi-e}{\chi}}. \end{eqnarray}$

(3.6)

The first-order rogue waves of Eq (3.1) is obtained by inserting (3.6) in (3.5) we get

$\begin{eqnarray} &&u = u_0+\frac{12\mu}{\delta} (\ln G_1)_{\zeta\zeta}, \end{eqnarray}$

(3.7)

where

$\begin{eqnarray} &&G_1 = {\frac {({d}^{2}\chi+\delta\, u_{{0}}+\chi-e)(z-\alpha)^2}{\chi}}+{(\zeta-\beta) }^{2}-\, {\frac {3\mu}{{d}^{2}\chi+\delta\, u_{{0}}+\chi-e}}. \end{eqnarray}$

(3.8)

shows first-order rogue wave solutions (3.7) at $\alpha = \beta = 0$ . These solutions have three centers $(0, 0)$ and $(\pm3\, \sqrt {-{\frac {\mu}{{d}^{2}\chi+\delta\, u_{{0}}+\chi-e}}}, 0)$ . In three-dimensional, contour plot and the corresponding density plot is presented. It is remarked that there is one peak, and the first-order rogue wave has the minimum amplitude $-{\frac {8\, {d}^{2}\chi+7\, \delta\, u_{{0}}+8\, \chi-8\, e}{\delta}}$ at $(0, 0)$ and maximal amplitude ${\frac {{d}^{2}\chi+2\, \delta\, u_{{0}}+\chi-e}{\delta}}$ at $(3\, \sqrt {-{\frac {\mu}{{d}^{2}\chi+\delta\, U_{{0}}+\chi-e}}}, 0)$ when $\mu > 0, {d}^{2}\chi+\delta\, u_{{0}}+\chi < e$ . The first-order rogue wave solutions (3.7) at $\alpha = -5, \beta = -5$ the center of rogue wave will be $(-5, -5)$ and $({\frac {-5\, {d}^{2}\chi-5\, \delta\, u_{{0}}+3\, \sqrt {-\chi\, {d}^{2}\mu -\delta\, \mu\, u_{{0}}-\chi\, \mu+e\mu}-5\, \chi+5\, e}{{d}^{2}\chi+\delta \, u_{{0}}+\chi-e}}, -5)$ as shown in , moreover, the minimal and maximal amplitudes also change into $-{\frac {8\, {d}^{2}\chi+7\, \delta\, u_{{0}}+8\, \chi-8\, e}{\delta}}$ and ${\frac {{d}^{2}\chi+2\, \delta\, u_{{0}}+\chi-e}{\delta}}$ respectively.

Figure 1. The first-order rogue wave solution (3.7). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. The first-order rogue wave solution (3.7). (d) 3D plot; (e) Contour plot; (f) Density at

$\alpha = \beta = -2$ .

DownLoad: Full-Size Img PowerPoint

3.2. Second-order rogue waves $n=1$

The second-order rogue wave solutions of Eq (3.1) can be found by setting $n = 1$ in Eq (2.5) as:

$\begin{eqnarray} &&F = G_2 \left( \zeta , z;\alpha, \beta \right) = 2\, \alpha\, z P_1 \left( \zeta , z\right) +F_2(\zeta, z)+ 2\, \beta\, \zeta \, Q_1 \left( \zeta , z \right) + \left( {\alpha}^{2}+{\beta}^{2} \right) F_0 \left( \zeta , z\right) = \\&& a_{{6, 0}}{\zeta }^{6}+ \left( {z}^{2}a_{{4, 2}}+a_{{4, 0}} \right) { \zeta }^{4}+2\, {\zeta }^{3}\beta\, c_{{2, 0}}+ \left( {z}^{4}a_{{2, 4}}+2 \, \alpha\, zb_{{2, 0}}+{z}^{2}a_{{2, 2}}+a_{{2, 0}} \right) {\zeta }^{2}+2 \, \beta\, \\&&\!\!\! \left( c_{{0, 2}}{z}^{2}+c_{{0, 0}} \right) \zeta +a_{{0, 6}}{z}^{6}+a_{{0, 4}}{z}^{4}+2\, \alpha\, {z}^{3}b_{{0, 2} }+a_{{0, 2}}{z}^{2}+2\, \alpha\, zb_{{0, 0}}+a_{{0, 0}} \left( {\alpha}^{2} +{\beta}^{2}+1 \right). \end{eqnarray}$

(3.9)

Substituting (3.9) in (3.4) and taking the coefficients of all powers of $\zeta$ and $z$ equal to zero, the set of parameters $a_{m, l}, b_{m, l}, c_{m.l}, (m, l = 0, 2, 4, 6)$ can be obtained as:

$\begin{eqnarray} &&a_{{0, 0}} = \frac {1}{9\, ( ( {d}^{2}+1 ) \chi+\delta \, u_{{0}}-e ) ^{3} ( {\alpha}^{2}+{\beta}^{2}+1 ) } ( 9\, ( {c_{{2, 0}}}^{2} ( {d}^{2}+1 ) {\beta}^{2 }+\frac{1}{9}\, {\alpha}^{2}{b_{{2, 0}}}^{2} ) ( {d}^{2}+1 ) ^ {2}{\chi}^{3}\\&&-27\, ( {c_{{2, 0}}}^{2} ( {d}^{2}+1 ) { \beta}^{2}+{\frac {2\, {\alpha}^{2}{b_{{2, 0}}}^{2}}{27}} ) ( {d}^{2}+1 ) ( -\delta\, u_{{0}}+e ) {\chi}^{2 }+27\, ( {c_{{2, 0}}}^{2} ( {d}^{2}+1 ) {\beta}^{2}+\frac{1}{27}\, {\alpha}^{2}{b_{{2, 0}}}^{2} )\\&& \times( -\delta\, u_{{0}}+e ) ^{2}\chi-9\, {c_{{2, 0}}}^{2} ( -\delta\, u_{{0}}+e ) ^{3}{\beta}^{2}-16875\, {\mu}^{3} ), a_{{0, 2}} = {\frac {{475\mu}^{2}}{\chi\, \left( {d}^{2}\chi+\delta\, u_ {{0}}+\chi-e \right) }}, \\&&a_{{0, 4}} = -\, {\frac {17\mu\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) }{{\chi}^{2}}}, a_{{0, 6}} = {\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^ {3}}{{\chi}^{3}}}, \\&&a_{{2, 0}} = -\, {\frac {{125\mu}^{2}}{ \left( {d}^{2}\chi+\delta\, u_{{0}} +\chi-e \right) ^{2}}} , a_{{2, 2}} = -\, {\frac {90\mu}{\chi}}, a_{{2, 4}} = \, {\frac {3 \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{2}}{{\chi}^{2}}}, \\&&a_{{4, 0}} = -\, {\frac {25\mu}{{d}^{2}\chi+\delta\, u_{{0}}+\chi-e}}, a_{{4, 2}} = {\frac {3\, {d}^{2}\chi+3\, \delta\, u_{{0}}+3\, \chi-3\, e}{\chi }}, \\&&b_{{0, 0}} = -\, {\frac {5\mu\, b_{{2, 0}}}{3\, {d}^{2}\chi+3\, \delta\, u_{{0} }+3\, \chi-3\, e}}, b_{{0, 2}} = -{\frac {b_{{2, 0}} \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) }{3\, \chi}}, \\&&c_{{0, 0}} = {\frac {\mu\, c_{{2, 0}}}{{d}^{2}\chi+\delta\, u_{{0}}+\chi-e}}, c_{{0, 2}} = -3\, {\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) c_{{2, 0}}}{\chi}}, \end{eqnarray}$

(3.10)

where $b_{{2, 0}}$ and $c_{2, 0}$ {are} arbitrary parameters.

The second-order rogue wave of Eq (3.1) can be found as:

$\begin{eqnarray} &&u = u_{{0}}+\frac{12\mu}{\delta} (\ln G_2 \left( \zeta , z;\alpha, \beta \right))_{\zeta\zeta}. \end{eqnarray}$

(3.11)

and show the two high peaks of the second-order rogue waves for (3.11) at $\alpha = \beta = 0$ . The second-order peak breaks apart and for sufficiently big parameters at $\alpha = \beta = 1000$ . The set of three first order rogue waves are forming a triangle called rogue wave triplet.

Figure 3. The second-order rogue wave solution (3.11). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. The second-order rogue wave solution (3.11). (d) 3D plot; (e) Contour plot; (f) Density at

$\alpha = \beta = 1000$ .

DownLoad: Full-Size Img PowerPoint

3.3. Third-order rogue waves $n=2$

The third-order rogue wave of Eq (3.1) is given by establishing $n = 2$ in Eq (2.5) as follows:

$\begin{eqnarray} &&F = G_3 \left( \zeta , z;\alpha, \beta \right) = 2\, \alpha\, z P_2 \left( \zeta , z\right) +F_3(\zeta, z)+ 2\, \beta\, \zeta \, Q_2 \left( \zeta , z \right) + \left( {\alpha}^{2}+{\beta}^{2} \right) F_1 \left( \zeta , z\right) \\&& = a_{{12, 0}}{\zeta }^{12}+a_{{10, 0}}{\zeta }^{10}+a_{{10, 2}}{z}^{2}{ \zeta }^{10}+a_{{8, 0}}{\zeta }^{8}+a_{{8, 2}}{z}^{2}{\zeta }^{8}+a_{{8, 4}}{z}^{4}{\zeta }^{8}+{\zeta }^{6}+a_{{6, 2}}{z}^{2}{\zeta }^{6}+a_{{6 , 4}}{z}^{4}{\zeta }^{6}\\&&+a_{{6, 6}}{z}^{6}{\zeta }^{6}+a_{{4, 0}}{\zeta } ^{4}+a_{{4, 2}}{z}^{2}{\zeta }^{4}+a_{{4, 4}}{z}^{4}{\zeta }^{4}+a_{{4, 6 }}{z}^{6}{\zeta }^{4}+a_{{4, 8}}{z}^{8}{\zeta }^{4}+a_{{2, 0}}{\zeta }^{ 2}+a_{{2, 2}}{z}^{2}{\zeta }^{2}\\&&+a_{{2, 4}}{z}^{4}{\zeta }^{2}+a_{{2, 6}} {z}^{6}{\zeta }^{2}+a_{{2, 8}}{z}^{8}{\zeta }^{2}+a_{{2, 10}}{z}^{10}{ \zeta }^{2}+2\, \beta\, ( c_{{6, 0}}{\zeta }^{6}+c_{{4, 2}}{z}^{2}{ \zeta }^{4}+c_{{4, 0}}{\zeta }^{4}+c_{{2, 4}}{z}^{4}{\zeta }^{2}\\&&+c_{{2, 2 }}{z}^{2}{\zeta }^{2}+c_{{2, 0}}{\zeta }^{2}+c_{{0, 6}}{z}^{6}+c_{{0, 4}} {z}^{4}+c_{{0, 2}}{z}^{2}+c_{{0, 0}} ) \left( \zeta \right) + \left( {\alpha}^{2}+{\beta}^{2} \right) ( a_{{2, 0}}{\zeta }^{2} +a_{{0, 2}}{z}^{2}\\&&+a_{{0, 0}} ) +a_{{0, 0}}+2\, \alpha\, z ( b_{ {6, 0}}{\zeta }^{6}+b_{{4, 2}}{z}^{2}{\zeta }^{4}+b_{{4, 0}}{\zeta }^{4}+ b_{{2, 4}}{z}^{4}{\zeta }^{2}+b_{{2, 2}}{z}^{2}{\zeta }^{2}+b_{{2, 0}}{ \zeta }^{2}+b_{{0, 6}}{z}^{6}\\&&+b_{{0, 4}}{z}^{4}+b_{{0, 2}}{z}^{2}+b_{{0, 0 }} ) +a_{{0, 2}}{z}^{2}+a_{{0, 4}}{z}^{4}+a_{{0, 6}}{z}^{6}+a_{{0, 8 }}{z}^{8}+a_{{0, 10}}{z}^{10}+a_{{0, 12}}{z}^{12}. \end{eqnarray}$

(3.12)

Substituting (3.12) in (3.4) and taken all coefficients of all powers of $\zeta$ and $z$ equal to zero, the set of parameters $a_{m, l}, b_{m, l}, c_{m.l}, (m, l = 0, 2, 4, 6)$ are found as:

$\begin{eqnarray} \label{e:B19} &&a_{{0, 0}} = \frac {1}{1863225\, ( ( {d}^{2}+1 ) \chi+\delta\, u_{{0}}-e ) ^{6}\mu\, ( {\alpha}^{2}+ {\beta}^{2}+1 ) } ( -33075\, ( {d}^{2}+1 ) ^{6}\\&& \times ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2}+{ \frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{11025}} ) {\chi}^{7}+ 231525\, ( -\delta\, u_{{0}}+e ) ( {d}^{2}+1 ) ^ {5} ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2 }\\&&+{\frac {338\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) {\chi}^{6} -694575\, ( -\delta\, u_{{0}}+e ) ^{2} ( {\beta}^{2}{d} ^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2}+{\frac {169\, {\alpha}^{ 2}{b_{{4, 0}}}^{2}}{15435}} ) ( {d}^{2}+1 ) ^{4}{\chi }^{5}\\&&+1157625\, ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c _{{4, 0}}}^{2}+{\frac {676\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) ( -\delta\, u_{{0}}+e ) ^{3} ( {d}^{2}+1 ) ^{3}{\chi}^{4}-1157625\, ( -\delta\, u_{{0}}\\&&+e ) ^{ 4} ( {d}^{2}+1 ) ^{2} ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^ {2}+{\beta}^{2}{c_{{4, 0}}}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2 }}{25725}} ) {\chi}^{3}+694575\, ( -\delta\, u_{{0}}+e ) ^{5} ( {d}^{2}+1 ) ( {\beta}^{2}{d}^{2}{c_{{ 4, 0}}}^{2}\\&&+{\beta}^{2}{c_{{4, 0}}}^{2}+{\frac {338\, {\alpha}^{2}{b_{{4, 0 }}}^{2}}{77175}} ) {\chi}^{2}-231525\, ( -\delta\, u_{{0}}+e ) ^{6} ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{ {4, 0}}}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) \chi\\&&-33075\, {\beta}^{2}{\delta}^{7}{u_{{0}}}^{7}{c_{{4, 0}}}^{2}+231525 \, {\beta}^{2}{\delta}^{6}e{u_{{0}}}^{6}{c_{{4, 0}}}^{2}-694575\, {\beta} ^{2}{\delta}^{5}{e}^{2}{u_{{0}}}^{5}{c_{{4, 0}}}^{2}+1157625\, {\beta}^{ 2}{\delta}^{4}{e}^{3}{u_{{0}}}^{4}{c_{{4, 0}}}^{2}\\&&-1157625\, {\beta}^{2} {\delta}^{3}{e}^{4}{u_{{0}}}^{3}{c_{{4, 0}}}^{2}+694575\, {\beta}^{2}{ \delta}^{2}{e}^{5}{u_{{0}}}^{2}{c_{{4, 0}}}^{2}-231525\, {\beta}^{2} \delta\, {e}^{6}u_{{0}}{c_{{4, 0}}}^{2}+33075\, {\beta}^{2}{e}^{7}{c_{{4, 0 }}}^{2}\\&&+181938957825625\, {\mu}^{7} ), \\&& a_{{0, 2}} = \frac {1}{ 1863225\, ( ( {d}^{2}+1 ) \chi+\delta\, u_{{0}}-e ) ^{4}{\mu}^{2}\chi\, ( {\alpha}^{2}+{\beta}^{2}+1 ) }\\&& \times( 11025\, ( {d}^{2}+1 ) ^{6} ( {\beta} ^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2}+{\frac {169\, { \alpha}^{2}{b_{{4, 0}}}^{2}}{11025}} ) {\chi}^{7}-77175\, ( -\delta\, u_{{0}}+e ) ( {d}^{2}+1 ) ^{5}\\&& \times( { \beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2}+{\frac { 338\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) {\chi}^{6}+231525\, ( -\delta\, u_{{0}}+e ) ^{2} ( {\beta}^{2}{d}^{2}{c_{{ 4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2}\\&&+{\frac {169\, {\alpha}^{2}{b_{{4, 0 }}}^{2}}{15435}} ) ({d}^{2}+1 ) ^{4}{\chi}^{5}- 385875\, ({\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0} }}^{2}+{\frac {676\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) ( -\delta\, u_{{0}}+e ) ^{3} ( {d}^{2}+1 ) ^{3}{ \chi}^{4}\\&&+385875\, ( -\delta\, u_{{0}}+e ) ^{4} ( {d}^{ 2}+1 ) ^{2} ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2} {c_{{4, 0}}}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) {\chi}^{3}\\ &&-231525\, ( -\delta\, u_{{0}}+e ) ^{5} ( {d}^{2}+1 ) ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{ \beta}^{2}{c_{{4, 0}}}^{2}+{\frac {338\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{ 77175}} ) {\chi}^{2}\\&&+77175\, ( -\delta\, u_{{0}}+e ) ^ {6} ({\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2 }+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) \chi+11025 \, {\beta}^{2}{\delta}^{7}{u_{{0}}}^{7}{c_{{4, 0}}}^{2}\\&&-77175\, {\beta}^{ 2}{\delta}^{6}e{u_{{0}}}^{6}{c_{{4, 0}}}^{2}+231525\, {\beta}^{2}{\delta }^{5}{e}^{2}{u_{{0}}}^{5}{c_{{4, 0}}}^{2}-385875\, {\beta}^{2}{\delta}^{ 4}{e}^{3}{u_{{0}}}^{4}{c_{{4, 0}}}^{2}+385875\, {\beta}^{2}{\delta}^{3}{ e}^{4}{u_{{0}}}^{3}{c_{{4, 0}}}^{2} \\ &&-231525\, {\beta}^{2}{\delta}^{2}{e}^ {5}{u_{{0}}}^{2}{c_{{4, 0}}}^{2}+77175\, {\beta}^{2}\delta\, {e}^{6}u_{{0 }}{c_{{4, 0}}}^{2}-11025\, {\beta}^{2}{e}^{7}{c_{{4, 0}}}^{2}- 186879449006250\, {\mu}^{7} ), \\&& a_{{0, 4}} = {\frac {16391725\, {\mu} ^{4}}{3\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{2}{\chi} ^{2}}}, a_{{0, 6}} = -{\frac {798980\, {\mu}^{3}}{3\, {\chi}^{3}}}, a_{{0, 8}} = 4335\, {\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{2} {\mu}^{2}}{{\chi}^{4}}}, \\&&a_{{0, 10}} = -58\, {\frac { \left( {d}^{2}\chi+ \delta\, u_{{0}}+\chi-e \right) ^{4}\mu}{{\chi}^{5}}}, a_{{0, 12}} = { \frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{6}}{{\chi} ^{6}}}, \\&&a_{{2, 0}} = \frac {1}{1863225\, ( ( {d}^{2}+1 ) \chi+\delta\, u_{{0}}-e ) ^{5}{\mu}^{2} ( {\alpha}^ {2}+{\beta}^{2}+1 ) }\\&& \times( 11025\, ( {d}^{2}+1 ) ^{ 6} ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2} +{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{11025}} ) {\chi}^{7}- 77175\, ( -\delta\, u_{{0}}+e ) ( {d}^{2}+1 ) ^{ 5} \\&&\times( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2} +{\frac {338\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) {\chi}^{6}+ 231525\, ( -\delta\, u_{{0}}+e ) ^{2} ( {\beta}^{2}{d}^ {2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2}\\&&+{\frac {169\, {\alpha}^{2 }{b_{{4, 0}}}^{2}}{15435}} ) ( {d}^{2}+1 ) ^{4}{\chi} ^{5}-385875\, ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{ {4, 0}}}^{2}+{\frac {676\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) ( -\delta\, u_{{0}}+e ) ^{3} \\&& \times( {d}^{2}+1 ) ^{3}{ \chi}^{4}+385875\, ( -\delta\, u_{{0}}+e ) ^{4} ( {d}^{ 2}+1 ) ^{2} ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2} {c_{{4, 0}}}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) {\chi}^{3}\\&&-231525\, ( -\delta\, u_{{0}}+e ) ^{5} ( {d}^{2}+1 ) ( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{ \beta}^{2}{c_{{4, 0}}}^{2}+{\frac {338\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{ 77175}} ) {\chi}^{2}+77175\, ( -\delta\, u_{{0}}+e ) ^ {6} \\&&\times( {\beta}^{2}{d}^{2}{c_{{4, 0}}}^{2}+{\beta}^{2}{c_{{4, 0}}}^{2 }+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) \chi+11025 \, {\beta}^{2}{\delta}^{7}{u_{{0}}}^{7}{c_{{4, 0}}}^{2}-77175\, {\beta}^{ 2}{\delta}^{6}e{u_{{0}}}^{6}{c_{{4, 0}}}^{2}\\&&+231525\, {\beta}^{2}{\delta }^{5}{e}^{2}{u_{{0}}}^{5}{c_{{4, 0}}}^{2}-385875\, {\beta}^{2}{\delta}^{ 4}{e}^{3}{u_{{0}}}^{4}{c_{{4, 0}}}^{2}+385875\, {\beta}^{2}{\delta}^{3}{ e}^{4}{u_{{0}}}^{3}{c_{{4, 0}}}^{2}\\&&-231525\, {\beta}^{2}{\delta}^{2}{e}^ {5}{u_{{0}}}^{2}{c_{{4, 0}}}^{2}+77175\, {\beta}^{2}\delta\, {e}^{6}u_{{0 }}{c_{{4, 0}}}^{2}-11025\, {\beta}^{2}{e}^{7}{c_{{4, 0}}}^{2}- 99239431541250\, {\mu}^{7} ), \\&&a_{{2, 2}} = 565950\, {\frac {{\mu}^{4 }}{ ( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e ) ^{3}\chi}}, a_{{2 , 4}} = 14700\, {\frac {{\mu}^{3}}{{\chi}^{2} ( {d}^{2}\chi+\delta\, u _{{0}}+\chi-e ) }}, \\&&a_{{2, 6}} = 35420\, {\frac { ( {d}^{2}\chi+ \delta\, u_{{0}}+\chi-e ) {\mu}^{2}}{{\chi}^{3}}}, a_{{2, 8}} = -570 \, {\frac { ( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e ) ^{3}\mu}{ {\chi}^{4}}}, \\&&a_{{2, 10}} = 6\, {\frac { \left( {d}^{2}\chi+\delta\, u_{{0}} +\chi-e \right) ^{5}}{{\chi}^{5}}}, a_{{4, 0}} = -{\frac {5187875\, {\mu}^{ 4}}{3\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{4}}}, \\&&a_{{4 , 2}} = -220500\, {\frac {{\mu}^{3}}{ \left( {d}^{2}\chi+\delta\, u_{{0}}+ \chi-e \right) ^{2}\chi}}, a_{{4, 4}} = 37450\, {\frac {{\mu}^{2}}{{\chi}^{ 2}}}, \\&&a_{{4, 6}} = -1460\, {\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi -e \right) ^{2}\mu}{{\chi}^{3}}}, a_{{4, 8}} = 15\, {\frac { \left( {d}^{2} \chi+\delta\, u_{{0}}+\chi-e \right) ^{4}}{{\chi}^{4}}}, \\&&a_{{6, 0}} = -{ \frac {75460\, {\mu}^{3}}{3\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{3}}}, a_{{6, 2}} = 18620\, {\frac {{\mu}^{2}}{\chi\, \left( {d}^ {2}\chi+\delta\, u_{{0}}+\chi-e \right) }}, \\&&a_{{6, 4}} = -1540\, {\frac {\mu \, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) }{{\chi}^{2}}}, a_ {{6, 6}} = 20\, {\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{3}}{{\chi}^{3}}}, \\&&a_{{8, 0}} = 735\, {\frac {{\mu}^{2}}{ \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{2}}}, a_{{8, 2}} = -690\, { \frac {\mu}{\chi}}, a_{{8, 4}} = 15\, {\frac { \left( {d}^{2}\chi+\delta\, u _{{0}}+\chi-e \right) ^{2}}{{\chi}^{2}}}, \\ &&a_{{10, 0}} = -98\, {\frac {\mu}{ {d}^{2}\chi+\delta\, u_{{0}}+\chi-e}}, a_{{10, 2}} = {\frac {6\, {d}^{2}\chi +6\, \delta\, u_{{0}}+6\, \chi-6\, e}{\chi}}, b_{{0, 0}} = {\frac {539\, b_{{4, 0 }}{\mu}^{2}}{9\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{2 }}}, \\&&b_{{0, 2}} = {\frac {7\, \mu\, b_{{4, 0}}}{3\, \chi}}, b_{{0, 4}} = -{\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{2}b_{{4, 0}}}{15\, {\chi}^{2}}}, b_{{0, 6}} = -{\frac { \left( {d}^{2}\chi+\delta\, u_{{0}}+ \chi-e \right) ^{4}b_{{4, 0}}}{105\, \mu\, {\chi}^{3}}}, \\&&b_{{2, 0}} = 19\, { \frac {\mu\, b_{{4, 0}}}{3\, {d}^{2}\chi+3\, \delta\, u_{{0}}+3\, \chi-3\, e} }, b_{{2, 2}} = -{\frac {38\, b_{{4, 0}} \left( {d}^{2}\chi+\delta\, u_{{0}}+ \chi-e \right) }{21\, \chi}}, \\&&b_{{2, 4}} = {\frac {3\, \left( {d}^{2}\chi+ \delta\, u_{{0}}+\chi-e \right) ^{3}b_{{4, 0}}}{35\, \mu\, {\chi}^{2}}}, b_{{4, 2}} = {\frac { \left( {d}^{2}\chi+\delta\, u_{{0} }+\chi-e \right) ^{2}b_{{4, 0}}}{21\, \chi\, \mu}}, \\&&b_{{6, 0}} = -{\frac {b_{ {4, 0}} \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) }{21\, \mu}}, c _{{0, 0}} = {\frac {12005\, c_{{4, 0}}{\mu}^{2}}{39\, \left( {d}^{2}\chi+ \delta\, u_{{0}}+\chi-e \right) ^{2}}}, \\&&c_{{0, 2}} = -{\frac {535\, \mu\, c_{ {4, 0}}}{13\, \chi}}, c_{{0, 4}} = {\frac {45\, \left( {d}^{2}\chi+\delta\, u _{{0}}+\chi-e \right) ^{2}c_{{4, 0}}}{13\, {\chi}^{2}}}, \\&&c_{{0, 6}} = -{ \frac {5\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) ^{4}c_{{4 , 0}}}{13\, \mu\, {\chi}^{3}}}, c_{{2, 0}} = 245\, {\frac {\mu\, c_{{4, 0}}}{13 \, {d}^{2}\chi+13\, \delta\, u_{{0}}+13\, \chi-13\, e}}, \\&&c_{{2, 2}} = -{\frac { 230\, c_{{4, 0}} \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right) }{13 \, \chi}}, c_{{2, 4}} = {\frac {5\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi -e \right) ^{3}c_{{4, 0}}}{13\, \mu\, {\chi}^{2}}}, \\&&c_ {{4, 2}} = {\frac {9\, \left( {d}^{2}\chi+\delta\, u_{{0}}+\chi-e \right)^{2}c_{{4, 0}}}{13\, \chi\, \mu}}, c_{{6, 0}} = -{\frac {c_{{4, 0}} \left( {d} ^{2}\chi+\delta\, u_{{0}}+\chi-e \right) }{13\, \mu}}, \end{eqnarray}$

(3.13)

where $b_{4, 0}$ and $c_{4, 0}$ are arbitrary constants.

Then the third-order rogue wave solution for Eq (3.1) is defined as:

$\begin{eqnarray} &&u = u_{{0}}+\frac{12\mu}{\delta} (\ln G_3 \left( \zeta , z;\alpha, \beta \right))_{\zeta\zeta}. \end{eqnarray}$

(3.14)

and show three high peaks of the third-order rogue waves for (3.14) at $\alpha = \beta = 0$ . The third-order peak breaks apart and for sufficiently big parameters at $\alpha = \beta = 10^{8}$ , the third-order rogue waves consist of five first-order rogue waves. These waves are located in the corners of a pentagon and the other sit in the center.

Figure 5. The third-order rogue wave solution (3.14). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. The third-order rogue wave solution (3.14). (d) 3D plot; (e) Contour plot; (f) Density at

$\alpha = \beta = 10^{8}$ .

DownLoad: Full-Size Img PowerPoint

4. The multi rogue waves for the second extended (3+1)-dimensional (KP) equation

The second extended (3+1)-dimensional (KP) equation ^[34] is:

$\begin{eqnarray} &&(u_{t}+\delta uu_x+\mu u_{xxx})_x+\chi (u_{xx}+u_{yy}+u_{zz})+\rho (u_{xy}+u_{yz}+u_{zx}) = 0, \end{eqnarray}$

(4.1)

where $\delta, \mu, \chi$ and $\rho$ are constants and $u$ is a wave amplitude functions in $x, y, z$ and $t$ .

The rogue-waves solutions for (4.1) can be obtained by finding the Hirota bilinear form by setting $\zeta = x+d y-et$ . Then, the ODE of (4.1) can be obtained as:

$\begin{eqnarray} &&\mu u_{\zeta\zeta\zeta\zeta}+(\delta u-e+2 \chi-\rho) u_{\zeta\zeta}+\delta u_\zeta^2+\chi u_{zz} = 0. \end{eqnarray}$

(4.2)

Using the following variable transformation

$\begin{eqnarray} &&u = u_0+\frac{12\mu}{\delta} (\ln F)_{\zeta\zeta}. \end{eqnarray}$

(4.3)

Then we can obtain the Hirota bilinear form for (4.1) by inserting (4.3) into (4.2) as

$\begin{eqnarray} &&(\mu D_{\zeta}^4+(\delta u_0-e+2 \chi-\rho) u_{\zeta\zeta}+\chi D_z^2)F\cdot F = 0. \end{eqnarray}$

(4.4)

The multi rogue wave solutions of the second extended (3+1)-dimensional KP equation (4.1) are given as Figures 7 and 8.

Figure 7. The first-order rogue wave solution (4.6). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. The first-order rogue wave solution (4.6). (d) 3D plot; (e) Contour plot; (f) Density at

$\alpha = \beta = -5$ .

DownLoad: Full-Size Img PowerPoint

4.1. Case 1. $n=0$

The coefficients $a_{0, 0}$ and $a_{0, 2}$ in Eq (3.5) can be give as follows

$\begin{eqnarray} &&a_{{0, 0}} = {\frac {3\mu}{-\delta\, u_{{0}}-2\, \chi+e+\rho}}, \qquad \qquad a_{{0, 2}} = {\frac {\delta\, u_{{0}}+2\, \chi-e-\rho}{\chi}} . \end{eqnarray}$

(4.5)

Inserting (4.5) into (3.5), the first-order rogue waves for Eq (4.1) can be obtained in the form

$\begin{eqnarray} &&u = u_0+\frac{12\mu}{\delta} (\ln F)_{\zeta\zeta}, \end{eqnarray}$

(4.6)

where

$\begin{eqnarray} &&F = {\frac {\delta\, u_{{0}}+2\, \chi-e-\rho}{\chi} (z-\alpha)^2}+{(\zeta-\beta) }^{2}+{\frac {3\mu}{-\delta\, u_{{0}}-2\, \chi+e+\rho}}. \end{eqnarray}$

(4.7)

The first-order rogue wave solutions (4.6) when $\alpha = \beta = 0$ are shown in . This figure has three centers $(0, 0)$ and $(\pm3\, \sqrt {-{\frac {\mu}{\delta\, u_{{0}}+2\, \chi-e-\rho}}}, 0)$ in three-dimensional, contour plot and the corresponding density plot. It is remarked that, there is one peak only because energy of the rogue wave is focused on the high peaks. The first-order rogue wave has the minimal amplitude ${\frac {-7\, \delta\, u_{{0}}-16\, \chi+8\, e+8\, \rho}{\delta}}$ at $(0, 0)$ and maximal amplitude ${\frac {2\, \delta\, u_{{0}}+2\, \chi-e-\rho}{\delta}}$ at $(\pm3\, \sqrt {-{\frac {\mu}{\delta\, u_{{0}}+2\, \chi-e-\rho}}}, 0)$ where $\mu > 0, \chi < \frac{1}{2}(-\delta\, u_{{0}}+e+\rho)$ . The first-order rogue wave solutions (4.6) at $\alpha = -5, \beta = -5$ with the centers of rogue wave will be at $(-5, -5)$ and $({\frac {5\, \delta\, u_{{0}}-5\, e+10\, \chi-5\, \rho-3\, \sqrt {\mu\, \left(-\delta\, u_{{0}}-2\, \chi+e+\rho \right) }}{-\delta\, u_{{0}}-2 \, \chi+e+\rho}}, -5)$ as shown in . The minimal and maximal amplitudes are changing into ${\frac {-7\, \delta\, u_{{0}}-16\, \chi+8\, e+8\, \rho}{\delta}}$ and ${\frac {2\, \delta\, u_{{0}}+2\, \chi-e-\rho}{\delta}}$ respectively.

4.2. Case 2. $n=1$

For this case the second-order rogue wave solutions of Eq (4.1) is:

$\begin{eqnarray} &&u = u_{{0}}+\frac{12\mu}{\delta} (\ln G_2 \left( \zeta , z;\alpha, \beta \right))_{\zeta\zeta}, \end{eqnarray}$

(4.8)

where $G_2 \left(\zeta, z; \alpha, \beta \right))$ is given by (3.9) with $n = 1$ and

$\begin{eqnarray} &&a_{{0, 0}} = \frac{1}{{ \left( 9\, {\alpha}^{2 }+9\, {\beta}^{2}+9 \right) \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}}} (9\, {c_{{2, 0}}}^{2} \left( -\delta\, u_{{0}}- 2\, \chi+e+\rho \right) ^{3}{\beta}^{2}-4\, {\alpha}^{2}{\chi}^{3}{b_{{2 , 0}}}^{2}\\&&\\&&+4\, {\alpha}^{2}{b_{{2, 0}}}^{2} \left( -\delta\, u_{{0}}+e+ \rho \right) {\chi}^{2}-{\alpha}^{2}{b_{{2, 0}}}^{2} \left( -\delta\, u_ {{0}}+e+\rho \right) ^{2}\chi+16875\, {\mu}^{3}), \\&&a_{{0, 2}} = -475\, {\frac {{\mu}^{2}}{\chi\, \left( - \delta\, u_{{0}}-2\, \chi+e+\rho \right) }}, a_{{0, 4}} = 17\, {\frac {\mu\, \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) }{{\chi}^{2}}}, \\&&a_{{0, 6 }} = -{\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}}{{ \chi}^{3}}}, a_{{2, 0}} = -125\, {\frac {{\mu}^{2}}{ \left( -\delta\, u_{{0} }-2\, \chi+e+\rho \right) ^{2}}}, a_{{2, 2}} = -90\, {\frac {\mu}{\chi}}, \\&&a_{ {2, 4}} = 3\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2} }{{\chi}^{2}}}, a_{{4, 0}} = 25\, {\frac {\mu}{-\delta\, u_{{0}}-2\, \chi+e+ \rho}}, a_{{4, 2}} = {\frac {3\, \delta\, u_{{0}}+6\, \chi-3\, e-3\, \rho}{\chi }}, \\&&b_{{0, 0}} = -\, {\frac {5\mu\, b_{{2, 0}}}{3\, \delta\, u_{{0}}+6\, \chi-3 \, e-3\, \rho}}, b_{{0, 2}} = {\frac {b_{{2, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) }{3\, \chi}}, c_{{0, 0}} = -{\frac {\mu\, c_{{2, 0}}}{-\delta\, u_{{0}}-2\, \chi+e+\rho}}, \\&&c_{{0, 2}} = 3\, { \frac {c_{{2, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) }{\chi }}, \end{eqnarray}$

(4.9)

where $b_{{2, 0}}$ and $c_{2, 0}$ is an arbitrary parameters.

In and , the two high peaks of the second-order rogue waves of (4.6) at $\alpha = \beta = 0$ are shown. At sufficiently big parameters, the set of three first order rogue waves forms and the centers are formed a triangle entitled a rogue wave triplet.

Figure 9. The second-order rogue wave solution (4.8). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 10. The second-order rogue wave solution (4.8). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 1000$ .

DownLoad: Full-Size Img PowerPoint

4.3. Case 3. $n=2$

The third-order rogue wave solutions for this case of Eq (4.1) can be obtained as follows

$\begin{eqnarray} &&u = u_{{0}}+\frac{12\mu}{\delta} (\ln G_3 \left( \zeta , z;\alpha, \beta \right))_{\zeta\zeta}, \end{eqnarray}$

(4.10)

where $G_3 \left(\zeta, z; \alpha, \beta \right))$ is given by (3.12) with $n = 2$ and

$\begin{eqnarray} &&a_{{0, 0}} = \frac {1}{1863225\, \mu\, ( {\alpha}^{2}+{ \beta}^{2}+1 ) ( -\delta\, u_{{0}}-2\, \chi+e+\rho ) ^ {6}} ( ( -32448\, {\alpha}^{2}{b_{{4, 0}}}^{2}-4233600\, {c_{{ 4, 0}}}^{2}{\beta}^{2} ) {\chi}^{7}\\&&+14817600\, ( {c_{{4, 0}}} ^{2}{\beta}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) ( -\delta\, u_{{0}}+e+\rho ) {\chi}^{6}-22226400\, ( -\delta\, u_{{0}}+e+\rho ) ^{2} ( {c_{{4, 0}}}^{2}{ \beta}^{2}\\&&+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{30870}} ) { \chi}^{5}+18522000\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{\frac {338\, { \alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) ( -\delta\, u_{{0}}+e +\rho ) ^{3}{\chi}^{4}-9261000\, ( {c_{{4, 0}}}^{2}{\beta}^{ 2}\\&&+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{51450}} ) ( - \delta\, u_{{0}}+e+\rho ) ^{4}{\chi}^{3}+2778300\, ( -\delta \, u_{{0}}+e+\rho ) ^{5} ( {c_{{4, 0}}}^{2}{\beta}^{2}+{ \frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) {\chi}^{2}\\&&- 463050\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{\frac {169\, {\alpha}^{2}{b _{{4, 0}}}^{2}}{154350}} ) ( -\delta\, u_{{0}}+e+\rho ) ^{6}\chi-33075\, {\beta}^{2}{\delta}^{7}{u_{{0}}}^{7}{c_{{4, 0} }}^{2}+231525\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{6}\\&&\times( e+\rho ) {u_{{0}}}^{6}-694575\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{5} ( e+\rho ) ^{2}{u_{{0}}}^{5}+1157625\, {c_{{4, 0}}}^{2}{ \beta}^{2}{\delta}^{4} ( e+\rho ) ^{3}{u_{{0}}}^{4}-1157625 \, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{3} \\&& \times( e+\rho ) ^{4}{u_ {{0}}}^{3}+694575\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{2} ( e+ \rho ) ^{5}{u_{{0}}}^{2}-231525\, {c_{{4, 0}}}^{2}{\beta}^{2} \delta\, ( e+\rho ) ^{6}u_{{0}}+33075\, {\beta}^{2}{e}^{7}{c _{{4, 0}}}^{2}\\&&+231525\, {\beta}^{2}{e}^{6}\rho\, {c_{{4, 0}}}^{2}+694575\, {\beta}^{2}{e}^{5}{\rho}^{2}{c_{{4, 0}}}^{2}+1157625\, {\beta}^{2}{e}^{4 }{\rho}^{3}{c_{{4, 0}}}^{2}+1157625\, {\beta}^{2}{e}^{3}{\rho}^{4}{c_{{4 , 0}}}^{2}\\&&+694575\, {\beta}^{2}{e}^{2}{\rho}^{5}{c_{{4, 0}}}^{2}+231525\, {\beta}^{2}e{\rho}^{6}{c_{{4, 0}}}^{2}+33075\, {\beta}^{2}{\rho}^{7}{c_{ {4, 0}}}^{2}+181938957825625\, {\mu}^{7} ) , \\&&a_{{0, 2}} = \frac {1}{ ( 1863225\, {\alpha}^{2}+1863225\, {\beta}^{2}+1863225 ) ( -\delta\, u_{{0}}-2\, \chi+e+\rho ) ^{4}{\mu}^{2}\chi} ( ( 10816\, {\alpha}^{2}{b_{{4, 0}}}^{2}\\&&+1411200\, {c_{{4, 0}} }^{2}{\beta}^{2} ) {\chi}^{7}-4939200\, ( {c_{{4, 0}}}^{2}{ \beta}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) ( -\delta\, u_{{0}}+e+\rho ) {\chi}^{6}+7408800\, ( - \delta\, u_{{0}}\\&&\\&&+e+\rho ) ^{2} ( {c_{{4, 0}}}^{2}{\beta}^{2}+ {\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{30870}} ) {\chi}^{5}- 6174000\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{\frac {338\, {\alpha}^{2}{ b_{{4, 0}}}^{2}}{77175}} ) ( -\delta\, u_{{0}}+e+\rho ) ^{3}{\chi}^{4}\\ &&+3087000\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{ \frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{51450}} ) ( - \delta\, u_{{0}}+e+\rho ) ^{4}{\chi}^{3}-926100\, ( -\delta \, u_{{0}}+e+\rho ) ^{5} ( {c_{{4, 0}}}^{2}{\beta}^{2}\\&&+{ \frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) {\chi}^{2}+ 154350\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{\frac {169\, {\alpha}^{2}{b _{{4, 0}}}^{2}}{154350}} ) ( -\delta\, u_{{0}}+e+\rho ) ^{6}\chi+11025\, {\beta}^{2}{\delta}^{7}{u_{{0}}}^{7}{c_{{4, 0} }}^{2} \\ && -77175\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{6} ( e+\rho ) {u_{{0}}}^{6}+231525\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{5} ( e+\rho ) ^{2}{u_{{0}}}^{5}-385875\, {c_{{4, 0}}}^{2}{\beta }^{2}{\delta}^{4} ( e+\rho ) ^{3}{u_{{0}}}^{4}\\&&+385875\, {c_{ {4, 0}}}^{2}{\beta}^{2}{\delta}^{3} ( e+\rho ) ^{4}{u_{{0}}} ^{3}-231525\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{2} ( e+\rho ) ^{5}{u_{{0}}}^{2}+77175\, {c_{{4, 0}}}^{2}{\beta}^{2}\delta\, ( e+\rho ) ^{6}u_{{0}}\\&&-11025\, {\beta}^{2}{e}^{7}{c_{{4, 0}} }^{2}-77175\, {\beta}^{2}{e}^{6}\rho\, {c_{{4, 0}}}^{2}-231525\, {\beta}^{ 2}{e}^{5}{\rho}^{2}{c_{{4, 0}}}^{2}-385875\, {\beta}^{2}{e}^{4}{\rho}^{3 }{c_{{4, 0}}}^{2}\\&&-385875\, {\beta}^{2}{e}^{3}{\rho}^{4}{c_{{4, 0}}}^{2}- 231525\, {\beta}^{2}{e}^{2}{\rho}^{5}{c_{{4, 0}}}^{2}-77175\, {\beta}^{2} e{\rho}^{6}{c_{{4, 0}}}^{2}-11025\, {\beta}^{2}{\rho}^{7}{c_{{4, 0}}}^{2} \\ &&-186879449006250\, {\mu}^{7} ), a_{{0, 4}} = {\frac {16391725\, {\mu }^{4}}{3\, {\chi}^{2} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{ 2}}}, \\&&a_{{0, 6}} = -{\frac {798980\, {\mu}^{3}}{3\, {\chi}^{3}}}, a_{{0, 8}} = 4335\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2}{\mu }^{2}}{{\chi}^{4}}}, \\&&a_{{0, 10}} = -58\, {\frac {\mu\, \left( -\delta\, u_{{0 }}-2\, \chi+e+\rho \right) ^{4}}{{\chi}^{5}}}, a_{{0, 12}} = {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{6}}{{\chi}^{6}}}, \\&&a_{ {2, 0}} = \frac {1}{ ( 1863225\, {\alpha}^{2}+1863225\, {\beta}^{2}+ 1863225 ) ( -\delta\, u_{{0}}-2\, \chi+e+\rho ) ^{5}{ \mu}^{2}} \\&&\times( ( -10816\, {\alpha}^{2}{b_{{4, 0}}}^{2}-1411200\, {c_{{4, 0}}}^{2}{\beta}^{2} ) {\chi}^{7}+4939200\, ( {c_{{4, 0 }}}^{2}{\beta}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{25725}} ) ( -\delta\, u_{{0}}+e+\rho ) {\chi}^{6}\\ &&-7408800\, ( -\delta\, u_{{0}}+e+\rho ) ^{2} ( {c_{{4, 0}}}^{2}{ \beta}^{2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{30870}} ) { \chi}^{5}+6174000\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{\frac {338\, { \alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} )\\&& \times ( -\delta\, u_{{0}}+e +\rho ) ^{3}{\chi}^{4}-3087000\, ( {c_{{4, 0}}}^{2}{\beta}^{ 2}+{\frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{51450}} ) ( - \delta\, u_{{0}}+e+\rho ) ^{4}{\chi}^{3}\\&&+926100\, ( -\delta \, u_{{0}}+e+\rho ) ^{5} ( {c_{{4, 0}}}^{2}{\beta}^{2}+{ \frac {169\, {\alpha}^{2}{b_{{4, 0}}}^{2}}{77175}} ) {\chi}^{2}- 154350\, ( {c_{{4, 0}}}^{2}{\beta}^{2}+{\frac {169\, {\alpha}^{2}{b _{{4, 0}}}^{2}}{154350}} ) \\ &&\times ( -\delta\, u_{{0}}+e+\rho ) ^{6}\chi-11025\, {\beta}^{2}{\delta}^{7}{u_{{0}}}^{7}{c_{{4, 0} }}^{2}+77175\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{6} ( e+\rho ) {u_{{0}}}^{6}-231525\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{5} \\&&\times ( e+\rho ) ^{2}{u_{{0}}}^{5}+385875\, {c_{{4, 0}}}^{2}{\beta }^{2}{\delta}^{4} ( e+\rho ) ^{3}{u_{{0}}}^{4}-385875\, {c_{ {4, 0}}}^{2}{\beta}^{2}{\delta}^{3} ( e+\rho ) ^{4}{u_{{0}}} ^{3}+231525\, {c_{{4, 0}}}^{2}{\beta}^{2}{\delta}^{2} \\ &&\times ( e+\rho ) ^{5}{u_{{0}}}^{2}-77175\, {c_{{4, 0}}}^{2}{\beta}^{2}\delta\, ( e+\rho ) ^{6}u_{{0}}+11025\, {\beta}^{2}{e}^{7}{c_{{4, 0}} }^{2}+77175\, {\beta}^{2}{e}^{6}\rho\, {c_{{4, 0}}}^{2}\\&&+231525\, {\beta}^{ 2}{e}^{5}{\rho}^{2}{c_{{4, 0}}}^{2}+385875\, {\beta}^{2}{e}^{4}{\rho}^{3 }{c_{{4, 0}}}^{2}+385875\, {\beta}^{2}{e}^{3}{\rho}^{4}{c_{{4, 0}}}^{2}+ 231525\, {\beta}^{2}{e}^{2}{\rho}^{5}{c_{{4, 0}}}^{2}\\&&+77175\, {\beta}^{2} e{\rho}^{6}{c_{{4, 0}}}^{2}+11025\, {\beta}^{2}{\rho}^{7}{c_{{4, 0}}}^{2} +99239431541250\, {\mu}^{7} ) , \\&&a_{{2, 2}} = -565950\, {\frac {{\mu}^ {4}}{\chi\, \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}}}, a_{{ 2, 4}} = -14700\, {\frac {{\mu}^{3}}{{\chi}^{2} \left( -\delta\, u_{{0}}-2 \, \chi+e+\rho \right) }}, \\&&a_{{2, 6}} = -35420\, {\frac {{\mu}^{2} \left( - \delta\, u_{{0}}-2\, \chi+e+\rho \right) }{{\chi}^{3}}}, a_{{2, 8}} = 570\, { \frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}\mu}{{\chi} ^{4}}}, \\&&a_{{2, 10}} = -6\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{5}}{{\chi}^{5}}}, a_{{4, 0}} = -{\frac {5187875\, {\mu}^{4}}{3\, \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{4}}}, \\&&a_{{4, 2}} = - 220500\, {\frac {{\mu}^{3}}{\chi\, \left( -\delta\, u_{{0}}-2\, \chi+e+ \rho \right) ^{2}}}, a_{{4, 4}} = 37450\, {\frac {{\mu}^{2}}{{\chi}^{2}}}, \\&&a _{{4, 6}} = -1460\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2}\mu}{{\chi}^{3}}}, a_{{4, 8}} = 15\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{4}}{{\chi}^{4}}}, \\&&a_{{6, 0}} = {\frac { 75460\, {\mu}^{3}}{3\, \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^ {3}}}, a_{{6, 2}} = -18620\, {\frac {{\mu}^{2}}{\chi\, \left( -\delta\, u_{{0 }}-2\, \chi+e+\rho \right) }}, \\&&a_{{6, 4}} = 1540\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) \mu}{{\chi}^{2}}}, a_{{6, 6}} = -20\, { \frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}}{{\chi}^{3 }}}, \\&&a_{{8, 0}} = 735\, {\frac {{\mu}^{2}}{ \left( -\delta\, u_{{0}}-2\, \chi +e+\rho \right) ^{2}}}, a_{{8, 2}} = -690\, {\frac {\mu}{\chi}}, a_{{8, 4}} = 15\, {\frac { \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2}}{{ \chi}^{2}}}, \\&&a_{{10, 0}} = 98\, {\frac {\mu}{-\delta\, u_{{0}}-2\, \chi+e+ \rho}}, a_{{10, 2}} = {\frac {6\, \delta\, u_{{0}}+12\, \chi-6\, e-6\, \rho}{ \chi}}, b_{{0, 0}} = {\frac {539\, {\mu}^{2}b_{{4, 0}}}{9\, \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2}}}, \\ &&b_{{0, 2}} = {\frac {7\, \mu\, b_{{4, 0 }}}{3\, \chi}}, b_{{0, 4}} = -{\frac {b_{{4, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2}}{15\, {\chi}^{2}}}, b_{{0, 6}} = -{\frac {b_{{4, 0} } \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{4}}{105\, \mu\, {\chi }^{3}}}, \\&&b_{{2, 0}} = 19\, {\frac {\mu\, b_{{4, 0}}}{3\, \delta\, u_{{0}}+6\, \chi-3\, e-3\, \rho}}, b_{{2, 2}} = {\frac {38\, b_{{4, 0}} \left( -\delta\, u_ {{0}}-2\, \chi+e+\rho \right) }{21\, \chi}}, \\&&b_{{2, 4}} = -{\frac {3\, b_{{4, 0 }} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}}{35\, \mu\, {\chi }^{2}}}, b_{{4, 2}} = {\frac {b_{{4, 0}} \left( -\delta \, u_{{0}}-2\, \chi+e+\rho \right) ^{2}}{21\, \mu\, \chi}}, \\&&b_{{6, 0}} = { \frac {b_{{4, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) }{21\, \mu}}, c_{{0, 0}} = {\frac {12005\, {\mu}^{2}c_{{4, 0}}}{39\, \left( -\delta \, u_{{0}}-2\, \chi+e+\rho \right) ^{2}}}, c_{{0, 2}} = -{\frac {535\, \mu\, c _{{4, 0}}}{13\, \chi}}, \\&&c_{{0, 4}} = {\frac {45\, c_{{4, 0}} \left( -\delta\, u _{{0}}-2\, \chi+e+\rho \right) ^{2}}{13\, {\chi}^{2}}}, c_{{0, 6}} = -{ \frac {5\, c_{{4, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{4 }}{13\, \mu\, {\chi}^{3}}}, \\&&c_{{2, 0}} = 245\, {\frac {\mu\, c_{{4, 0}}}{13\, \delta\, u_{{0}}+26\, \chi-13\, e-13\, \rho}}, c_{{2, 2}} = {\frac {230\, c_{{4 , 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) }{13\, \chi}}, \\&&c_{{2 , 4}} = -{\frac {5\, c_{{4, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{3}}{13\, \mu\, {\chi}^{2}}}, c_{{4, 2}} = { \frac {9\, c_{{4, 0}} \left( -\delta\, u_{{0}}-2\, \chi+e+\rho \right) ^{2 }}{13\, \mu\, \chi}}, \\&&c_{{6, 0}} = {\frac {c_{{4, 0}} \left( -\delta\, u_{{0}} -2\, \chi+e+\rho \right) }{13\, \mu}}, \end{eqnarray}$

(4.11)

where $b_{4, 0}$ and $c_{4, 0}$ are arbitrary constants.

In and , the three high peaks of the third-order rogue waves for (4.6) at $\alpha = \beta = 0$ are introduced. The third-order peak breaks apart and for sufficiently big parameters for $\alpha = \beta = 10^{8}$ , the third-order rogue waves consists of five first-order rogue waves are located in the corners of a pentagon and other one sites in the center.

Figure 11. The third-order rogue wave solution (4.10). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 12. The third-order rogue wave solution (4.10). (a) 3D plot; (b) Contour plot; (c) Density at

$\alpha = \beta = 10^{8}$ .

DownLoad: Full-Size Img PowerPoint

5. Conclusions

In this paper, we investigated the first, second, and third-order rogue waves for two (3+1)-dimensional extensions of the (KP) equation by the bilinear method via the symbolic computation approach. The properties of the two (3+1)-dimensional extensions of the (KP) equation are examined by introducing several figures. The obtained higher-order rogue waves have the property $\lim_{x\rightarrow \pm\infty} = \lim_{y\rightarrow \pm\infty} = \lim_{z\rightarrow \pm\infty} = \lim_{t\rightarrow \pm\infty} = u_0$ . The figures were depicted in three dimensional, contour and density with the center controlled by the parameters $\alpha$ and $\beta$ . The results obtained in this work are useful to understand the dynamic behaviors of higher-rogue waves in the deep ocean and nonlinear optical fibers. Thus, the characteristics of these solutions are discussed through some diverting graphics under different parameter choices. The dynamics behaviors of higher-rogue waves related to the optical rogue waves are pulses of light similar to rogue or freak ocean waves. Rogue waves in optical fibers can be described mathematically by the nonlinear Schr $\ddot{o}$ dinger equation and its extensions that take into account third-order dispersion. We can apply this technique to completely integrable nonlinear evolution equations.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP. 2/36/43). This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R229), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

All authors declare that they have no conflicts of interest.

References

[1]	W. P. M. H. Heemels, K. H. Johansson, P. Tabuada, An introduction to event-triggered and self-triggered control, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012, 3270–3285. https://doi.org/10.1109/CDC.2012.6425820
[2]	Y. Wang, L. Xiao, Y. Guo, Finite-time stability of singular switched systems with a time-varying delay based on an evnet-triggered mechanism, AIMS Math., 8 (2023), 1901–1924. https://doi.org/10.3934/math.2023098 doi: 10.3934/math.2023098
[3]	L. Cao, Y. Pan, H. Liang, T. Huang, Observer-based dynamic event-triggered control for multiagent systems with time-varying delay, IEEE Trans. Cybern., 53 (2022), 3376–3387. https://doi.org/10.1109/TCYB.2022.3226873 doi: 10.1109/TCYB.2022.3226873
[4]	L. Cao, Z. Cheng, Y. Liu, H. Li, Event-based adaptive NN fixed-time cooperative formation for multiagent systems, IEEE Trans. Neural Networks Learn. Syst., 2022. https://doi.org/10.1109/TNNLS.2022.3210269
[5]	H. Yu, F. Hao, Periodic event-triggered state-feedback control for discrete-time linear systems, J. Franklin Inst., 353 (2016), 1809–1828. https://doi.org/10.1016/j.jfranklin.2016.03.002 doi: 10.1016/j.jfranklin.2016.03.002
[6]	W. Wu, S. Reimann, D. Görges, S. Liu, Event-triggered control for discrete-time linear systems subject to bounded disturbance, Int. J. Robust Nonlinear Control, 26 (2016), 1902–1918. https://doi.org/10.1002/rnc.3388 doi: 10.1002/rnc.3388
[7]	A. Eqtami, D. V. Dimarogonas, K. J. Kyriakopoulos, Event-triggered control for discrete-time systems, Proceedings of the 2010 American Control Conference, 2010, 4719–4724. https://doi.org/10.1109/ACC.2010.5531089
[8]	S. Ding, X. Xie, Y. Liu, Event-triggered static/dynamic feedback control for discrete-time linear systems, Inf. Sci., 524 (2020), 33–45. https://doi.org/10.1016/j.ins.2020.03.044 doi: 10.1016/j.ins.2020.03.044
[9]	P. H. S. Coutinho, M. L. C. Peixoto, I. Bessa, R. M. Palhares, Dynamic event-triggered gain-scheduling control of discrete-time quasi-LPV systems, Automatica, 141 (2022), 110292. https://doi.org/10.1016/j.automatica.2022.110292 doi: 10.1016/j.automatica.2022.110292
[10]	W. P. M. H. Heemels, M. C. F. Donkers, Model-based periodic event-triggered control for linear systems, Automatica, 49 (2013), 698–711. https://doi.org/10.1016/j.automatica.2012.11.025 doi: 10.1016/j.automatica.2012.11.025
[11]	Y. Zhang, J. Wang, Y. Xu, A dual neural network for bi-criteria kinematic control of redundant manipulators, IEEE Trans. Robot. Autom., 18 (2002), 923–931. https://doi.org/10.1109/TRA.2002.805651 doi: 10.1109/TRA.2002.805651
[12]	Y. Oh, M. H. Lee, J. Moon, Infinity-norm-based worst-case collision avoidance control for quadrotors, IEEE Access, 9 (2021), 101052–101064. https://doi.org/10.1109/ACCESS.2021.3096275 doi: 10.1109/ACCESS.2021.3096275
[13]	J. Imura, A. van der Schaft, Characterization of well-posedness of piecewise-linear systems, IEEE Trans. Autom. Control, 45 (2000), 1600–1619. https://doi.org/10.1109/9.880612 doi: 10.1109/9.880612
[14]	G. Ferrari-Trecate, F. A. Cuzzola, D. Mignone, M. Morari, Analysis of discrete-time piecewise affine and hybrid systems, Automatica, 38 (2002), 2139–2146. https://doi.org/10.1016/S0005-1098(02)00142-5 doi: 10.1016/S0005-1098(02)00142-5
[15]	J. Doyle, K. Glover, P. Khargonekar, B. Francis, State-space solutions to standard $H_2$ and $H_\infty$ control problems, 1988 American Control Conference, 1988, 1691–1696. https://doi.org/10.23919/ACC.1988.4789992
[16]	C. E. Souza, L. Xie, On the discrete-time bounded real lemma with application in the characterization of static state feedback $H_\infty$ controllers, Syst. Control Lett., 18 (1992), 61–71. https://doi.org/10.1016/0167-6911(92)90108-5 doi: 10.1016/0167-6911(92)90108-5
[17]	S. Luemsai, T. Botmart, W. Weera, S. Charoensin, Improved results on mixed passive and $H_{\infty}$ performance for uncertain neural networks with mixed interval time-varying delays via feedback control, AIMS Math., 6 (2021), 2653–2679. https://doi.org/10.3934/math.2021161 doi: 10.3934/math.2021161
[18]	D. A. Wilson, M. A. Nekoui, G. D. Halikias, An LQR weight selection approach to the discrete generalized $H_2$ control problem, Int. J. Control, 71 (1998), 93–101. https://doi.org/10.1080/002071798221948 doi: 10.1080/002071798221948
[19]	J. H. Kim, T. Hagiwara, Upper/lower bounds of generalized $H_2$ norms in sampled-data systems with convergence rate analysis and discretization viewpoint, Syst. Control. Lett., 107 (2017), 28–35. https://doi.org/10.1016/j.sysconle.2017.06.008 doi: 10.1016/j.sysconle.2017.06.008
[20]	J. H. Kim, T. Hagiwara, The generalized $H_2$ controller synthesis problem of sampled-data systems, Automatica, 142 (2022), 110400. https://doi.org/10.1016/j.automatica.2022.110400 doi: 10.1016/j.automatica.2022.110400
[21]	J. H. Kim, T. Hagiwara, $L_1$ discretization for sampled-data controller synthesis via piecewise linear approximation, IEEE Trans. Autom. Control, 61 (2016), 1143–1157. https://doi.org/10.1109/TAC.2015.2452815 doi: 10.1109/TAC.2015.2452815
[22]	J. H. Kim, T. Hagiwara, $L_1$ optimal controller synthesis for sampled-data systems via piecewise linear kernel approximation, Int. J. Robust Nonlinear Control, 31 (2021), 4933–4950. https://doi.org/10.1002/rnc.5513 doi: 10.1002/rnc.5513
[23]	W. P. M. H. Heemels, M. C. F. Donkers, A. R. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Autom. Control, 58 (2013), 847–861. https://doi.org/10.1109/TAC.2012.2220443 doi: 10.1109/TAC.2012.2220443
[24]	T. Chen, B. A. Francis, Optimal sampled-data control systems, London: Springer, 1995. https://doi.org/10.1007/978-1-4471-3037-6

This article has been cited by:

1.	Yong Wang, Yani Li, Qian Lu, Jiamin Zhang, Xiaoyu Zhang, Ziruo Ding, Huiyi Xu, Research on Evolutionary Game of Pharmaceutical Supply Chain Platform Considering Government Supervision Behavior, 2024, 12, 2169-3536, 143930, 10.1109/ACCESS.2024.3471643
2.	Zhifei Chen, Jun Wan, Fangkun Liu, Zhiping Hou, Driving Enterprise Digital Transformation: Unveiling the Significance of E-Commerce Demonstration City in China, 2024, 71, 0018-9391, 5641, 10.1109/TEM.2024.3366450
3.	Fan Lv, Research on optimization strategies of university ideological and political parenting models under the empowerment of digital intelligence, 2025, 15, 2045-2322, 10.1038/s41598-025-92985-8
4.	Fangfang Song, The effects of digital transformation on corporate energy efficiency: a supply chain spillover perspective, 2025, 6, 2673-4524, 10.3389/frsus.2025.1567413
5.	Lei Wang, Zixin Peng, Jiayi Ren, Linyuan Pang, Research on the identification of key factors and influence mechanism of digital transformation of manufacturing enterprises based on machine learning, 2025, 0167-2533, 10.1177/01672533251339606
6.	Huan Hu, Wenming Zhang, Influencing factors and realistic path of digital and intelligent transformation in manufacturing enterprises: a dynamic QCA analysis based on the TOE framework, 2025, 0953-7325, 1, 10.1080/09537325.2025.2503424

Reader Comments

Your name:*

Email:*
© 2023 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 Mathematics

1.8 3.4

Metrics

Article views(1588) PDF downloads(108) Cited by(6)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(4) / Tables(3)

AIMS Mathematics

Output-based event-triggered control for discrete-time systems with three types of performance analysis

Related Papers:

Abstract

1. Introduction

2. Materials and methods

3. The bilinear form and multi rogue waves for the first extended (3+1)-dimensional KP equation

3.1. First-order rogue waves $n=0$

3.2. Second-order rogue waves $n=1$

3.3. Third-order rogue waves $n=2$

4. The multi rogue waves for the second extended (3+1)-dimensional (KP) equation

4.1. Case 1. $n=0$

4.2. Case 2. $n=1$

4.3. Case 3. $n=2$

5. Conclusions

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

Catalog

AIMS Mathematics

Output-based event-triggered control for discrete-time systems with three types of performance analysis

Related Papers:

Abstract

1. Introduction

2. Materials and methods

3. The bilinear form and multi rogue waves for the first extended (3+1)-dimensional KP equation

3.1. First-order rogue waves n=0 n=0

3.2. Second-order rogue waves n=1 n=1

3.3. Third-order rogue waves n=2 n=2

4. The multi rogue waves for the second extended (3+1)-dimensional (KP) equation

4.1. Case 1. n=0 n=0

4.2. Case 2. n=1 n=1

4.3. Case 3. n=2 n=2

5. Conclusions

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

3.1. First-order rogue waves $n=0$

3.2. Second-order rogue waves $n=1$

3.3. Third-order rogue waves $n=2$

4.1. Case 1. $n=0$

4.2. Case 2. $n=1$

4.3. Case 3. $n=2$