Synchronization analysis of coupled fractional-order neural networks with time-varying delays

Biwen Li; Xuan Cheng; Biwen Li; Xuan Cheng

doi:10.3934/mbe.2023665

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 14846-14865. doi: 10.3934/mbe.2023665

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Research article

Synchronization analysis of coupled fractional-order neural networks with time-varying delays

Biwen Li ,
Xuan Cheng ^,

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received: 07 April 2023 Revised: 13 June 2023 Accepted: 28 June 2023 Published: 10 July 2023

In this paper, the complete synchronization and Mittag-Leffler synchronization problems of a kind of coupled fractional-order neural networks with time-varying delays are introduced and studied. First, the sufficient conditions for a controlled system to reach complete synchronization are established by using the Kronecker product technique and Lyapunov direct method under pinning control. Here the pinning controller only needs to control part of the nodes, which can save more resources. To make the system achieve complete synchronization, only the error system is stable. Next, a new adaptive feedback controller is designed, which combines the Razumikhin-type method and Mittag-Leffler stability theory to make the controlled system realize Mittag-Leffler synchronization. The controller has time delays, and the calculation can be simplified by constructing an appropriate auxiliary function. Finally, two numerical examples are given. The simulation process shows that the conditions of the main theorems are not difficult to obtain, and the simulation results confirm the feasibility of the theorems.

Keywords:

Citation: Biwen Li, Xuan Cheng. Synchronization analysis of coupled fractional-order neural networks with time-varying delays[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 14846-14865. doi: 10.3934/mbe.2023665

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Abstract

1. Introduction

Nonlinear partial differential equation is a very important branch of the nonlinear science, which has been called the foreword and hot topic of current scientific development. In theoretical science and practical application, nonlinear partial differential is used to describe the problems in the fields of optics, mechanics, communication, control science and biology ^{[1,2,3,4,5,6,7,8,9]}. At present, the main problems in the study of nonlinear partial differential equations are the existence of solutions, the stability of solutions, numerical solutions and exact solutions. With the development of research, especially the study of exact solutions of nonlinear partial differential equations has important theoretical value and application value. In the last half century, many important methods for constructing exact solutions of nonlinear partial differential equations have been proposed, such as the planar dynamic system method ^[10], the Jacobi elliptic function method ^[11], the bilinear transformation method ^[12], the complete discriminant system method for polynomials ^[13], the unified Riccati equation method ^[14], the generalized Kudryashov method ^[15], and so on ^{[16,17,18,19,20,21,22,23,24]}.

There is no unified method to obtain the exact solution of nonlinear partial differential equations. Although predecessors have obtained some analytical solutions with different methods, no scholar has studied the system with complete discrimination system for polynomial method.

The Fokas system is a very important class of nonlinear partial differential equations. In this article, we focus on the Fokas system, which is given as follows ^{[25,26,27,28,29,30,31,32,33,34,35,36,37]}

$\left\{ \begin{array}{l} i{p_t} + {r_1}{p_{xx}} + {r_2}pq = 0, \hfill \\ {r_3}{q_y} - {r_4}{(|p{|^2})_x} = 0, \hfill \end{array} \right.$

(1.1)

where $p = p(x, y, t)$ and $q = q(x, y, t)$ are the complex functions which stand for the nonlinear pulse propagation in monomode optical fibers. The parameters ${r_1}, {r_2}, {r_3}$ and ${r_4}$ are arbitrary non-zero constants, which are coefficients of nonlinear terms in Eq (1.1) and reflect different states of optical solitons.

This paper is arranged as follows. In Section 2, we describe the method of the complete discrimination system for polynomial method. In Section 3, we substitute traveling wave transformation into nonlinear ordinary differential equations and obtain the different new single traveling wave solutions for the Fokas system by complete discrimination system for polynomial method. At the same time, we draw some images of solutions. In Section 4, the main results are summarized.

2. Description of the method

First, we consider the following partial differential equations:

$\left\{ \begin{array}{l}F(u, v, {u}_{x}, {u}_{t}, {v}_{x}, {v}_{t}, {u}_{xx}, {u}_{xt}, {u}_{tt}, \cdots ) = 0\text{, }\\ G(u, v, {u}_{x}, {u}_{t}, {v}_{x}, {v}_{t}, {u}_{xx}, {u}_{xt}, {u}_{tt}, \cdots ) = 0\text{, }\end{array} \right.$

(2.1)

where $F$ and $G$ is polynomial function which is about the partial derivatives of each order of $u(x, t)$ and $v(x, t)$ with respect to $x$ and $t$ .

Step 1: Taking the traveling wave transformation $u(x, t) = u(\xi), v(x, t) = v(\xi), \xi = kx + ct$ into Eq (2.1), then the partial differential equation is converted to an ordinary differential equation

$\left\{ \begin{array}{l} F(u, v, u', v', u'', v'', \cdots ) = 0, \hfill \\ G(u, v, u', v', u'', v'', \cdots ) = 0. \hfill \end{array} \right.$

(2.2)

Step 2: The above nonlinear ordinary differential equations (2.2) are reduced to the following ordinary differential form after a series of transformations:

${(u')^2} = {u^3} + {d_2}{u^2} + {d_1}u + {d_0}.$

(2.3)

The Eq (2.3) can also be written in integral form as:

$\pm (\xi - {\xi _0}) = \int {\frac{{du}}{{\sqrt {{u^3} + {d_2}{u^2} + {d_1}u + {d_0}} }}} .$

(2.4)

Step 3: Let $\phi (u) = {u^3} + {d_2}{u^2} + {d_1}u + {d_0}$ . According to the complete discriminant system method of third-order polynomial

$\left\{\begin{array}{l} \Delta=-27(\frac{2 d_2^3}{27}+d_0-\frac{d_1 d_2}{3})^2-4(d_1-\frac{d_2^2}{3})^3, \\ D_1=d_1-\frac{d_2^2}{3}, \end{array}\right.$

(2.5)

the classification of the solution of the equation can be obtained, and the classification of traveling wave solution of the Fokas system will be given in the following section.

3. Traveling wave solution of Eq (1.1)

In the current part, we obtain all exact solutions to Eq (1.1) by complete discrimination system for polynomial method. According to the wave transformation

$p(x, y, t) = \varphi (\eta ){e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, q(x, y, t) = \phi (\eta ), \eta = x + y - vt,$

(3.1)

where ${\lambda _1}, {\lambda _2}, {\lambda _3}, {\lambda _4}$ and $v$ are real parameters, and $v$ represents the wave frame speed.

Substituting the above transformation Eq (3.1) into Eq (1.1), we get

$\left\{ \begin{array}{l} ( - v + 2{r_1}{\lambda _1})i\varphi ' - {\lambda _3}\varphi + {r_1}\varphi '' - {r_1}\lambda _1^2\varphi + {r_2}\varphi \phi = 0, \hfill \\ {r_3}\phi ' - 2{r_4}\varphi \varphi ' = 0. \hfill \end{array} \right.$

(3.2)

Integrating the second equation in (3.2) and ignoring the integral constant, we get

$\phi (\eta ) = \frac{{{r_4}{\varphi ^2}(\eta )}}{{{r_3}}}.$

(3.3)

Substituting Eq (3.3) into the first equation in (3.2) and setting $v = 2{r_1}{\lambda _1},$ we get the following:

${r_1}\varphi '' - ({\lambda _3} + {r_1}\lambda _1^2)\varphi + \frac{{{r_2}{r_4}{\varphi ^3}}}{{{r_3}}} = 0.$

(3.4)

Multiplying $\varphi '$ both sides of the Eq (3.4), then integrating once to get

${(\varphi ')^2} = {a_4}{\varphi ^4} + {a_2}{\varphi ^2} + {a_0},$

(3.5)

where ${a_4} = - \frac{{{r_2}{r_4}}}{{2{r_1}{r_3}}}, {a_2} = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_1}}},$ ${a_0}$ is the arbitrary constant.

${\rm{Let}} \ \ \varphi = \pm \sqrt {{{(4{a_4})}^{ - \frac{1}{3}}}\omega } , \ {b_1} = 4{a_2}{(4{a_4})^{ - \frac{2}{3}}}, {b_0} = 4{a_0}{(4{a_4})^{ - \frac{1}{3}}}, {\eta _1} = {(4{a_4})^{\frac{1}{3}}}\eta .$

(3.6)

Equation (3.5) can be expressed as the following:

${({\omega '_{{\eta _1}}})^2} = {\omega ^3} + {b_1}{\omega ^2} + {b_0}\omega .$

(3.7)

Then we can get the integral expression of Eq (3.7)

$\pm ({\eta _1} - {\eta _0}) = \int {\frac{{d\omega }}{{\sqrt {\omega ({\omega ^2} + {b_1}\omega + {b_0})} }}} ,$

(3.8)

where ${\eta _0}$ is the constant of integration.

Here, we get the $F(\omega) = {\omega ^2} + {b_1}\omega + {b_0}$ and $\Delta = b_1^2 - 4{b_0}.$ In order to solve Eq (3.7), we discuss the third order polynomial discrimination system in four cases.

Case 1: $\Delta = 0$ and $\omega \gt 0.$

When ${b_1} \lt 0,$ the solution of Eq (3.7) is

${\omega _1} = - \frac{{{b_1}}}{2}{\tanh ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.9)

${\omega _2} = - \frac{{{b_1}}}{2}{\coth ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.10)

Thus, the classification of all solutions of Eq (3.7) is obtained by the third order polynomial discrimination system. The exact traveling wave solutions of the Eq (1.1) are obtained by combining the above solutions and the conditions (3.6) with Eq (3.1), can be expressed as below:

${p_1}(x, y, t) = \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \tanh (\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.11)

In Eq (3.11), ${p_1}(x, y, t)$ is a dark soliton solution, it expresses the energy depression on a certain intensity background. Figure 1 depict two-dimensional graph, three-dimensional graph, contour plot and density plot of the solution.

${q_1}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tanh ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0}))$

(3.12)

$\begin{array}{l} {p_2}(x, y, t) = \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \coth (\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot \\ ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.13)

Figure 1. Module length graphs of Eq (3.12) when

${r_1} = - 2, {r_2} = 1, {r_3} = - 1, {r_4} = 1, {\lambda _1} = - 1, {\lambda _3} = 3, {\eta _0} = 0$ .

DownLoad: Full-Size Img PowerPoint

where ${p_1}(x, y, t), {q_1}(x, y, t), {p_2}(x, y, t), {q_2}(x, y, t)$ are hyperbolic function solutions. Specially, ${p_2}(x, y, t)$ is a bright soliton solution.

${q_2}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\coth ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})).$

(3.14)

When ${b_1} \gt 0,$ the solution of Eq (3.7) is

${\omega _3} = \frac{{{b_1}}}{2}{\tan ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})).$

(3.15)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_3}(x, y, t) = \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}} \tan (\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot \\ ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}. \end{array}$

(3.16)

${q_3}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tan ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})).$

(3.17)

In Eq (3.16) and Eq (3.17), ${p_3}(x, y, t)$ and ${q_3}(x, y, t)$ are trigonometric function solutions. ${q_3}(x, y, t)$ is a periodic wave solution, and it Shows the periodicity of ${q_3}(x, y, t)$ in Figure 2(a), (b).

Figure 2. Module length graphs of Eq (3.17) when

${r_1} = 2, {r_2} = 1, {r_3} = 1, {r_4} = 1, {\lambda _1} = 1, {\lambda _3} = - 1, {\eta _0} = 0$ .

DownLoad: Full-Size Img PowerPoint

When ${b_1} = 0,$ the solution of Eq (3.7) is

${\omega _4} = \frac{4}{{{{({\eta _1} - {\eta _0})}^2}}}.$

(3.18)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_4}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{ - \frac{1}{3}}}} \frac{2}{{{{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}}}{e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}},$

(3.19)

${q_4}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{ - \frac{1}{3}}}\frac{4}{{{{({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})}^2}}},$

(3.20)

where ${p_4}(x, y, t)$ is exponential function solution, and ${q_4}(x, y, t)$ is rational function solution.

Case 2: $\Delta = 0$ and ${b_0} = 0.$

When $\omega \gt - {b_1}$ and ${b_1} \lt 0,$ the solution of Eq (3.7) is

${\omega _5} = \frac{{{b_1}}}{2}{\tanh ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.21)

${\omega _6} = \frac{{{b_1}}}{2}{\coth ^2}(\frac{1}{2}\sqrt {\frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.22)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_5}(x, y, t) = \\ \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\tanh }^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) - 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.23)

${q_5}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tanh ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.24)

$\begin{array}{l} {p_6}(x, y, t) = \\ \pm \sqrt { - \frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\coth }^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) - 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.25)

${q_6}(x, y, t) = - \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\coth ^2}(\frac{1}{2}\sqrt {\frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.26)

where ${p_5}(x, y, t), {q_5}(x, y, t), {p_6}(x, y, t)$ and ${q_6}(x, y, t)$ are hyperbolic function solutions.

When $\omega \gt - {b_1}$ and ${b_1} \gt 0,$ the solution of Eq (3.7) is

${\omega _7} = - \frac{{{b_1}}}{2}{\tan ^2}(\frac{1}{2}\sqrt { - \frac{{{b_1}}}{2}} ({\eta _1} - {\eta _0})) - {b_1}.$

(3.27)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

$\begin{array}{l} {p_7}(x, y, t) = \\ \pm \sqrt {\frac{{{r_3}({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}{r_4}}}({{\tan }^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0})) + 2)} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}, \end{array}$

(3.28)

${q_7}(x, y, t) = \frac{{{\lambda _3} + {r_1}\lambda _1^2}}{{{r_2}}}{\tan ^2}(\frac{1}{2}\sqrt { - \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_1}}} \cdot {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{2}{3}}}}} \cdot ({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0})) + \frac{{2({\lambda _3} + {r_1}\lambda _1^2)}}{{{r_2}}},$

(3.29)

where ${p_7}(x, y, t)$ and ${q_7}(x, y, t)$ are trigonometric function solutions.

Case 3: $\Delta \gt 0$ and ${b_0} \ne 0.$ Let $u \lt v \lt s,$ there $u, v$ and $s$ are constants satisfying one of them is zero and two others are the root of $F(\omega) = 0.$

When $u \lt \omega \lt v,$ we can get the solution of Eq (3.7) is

${\omega _8} = u + (v - u)s{n^2}(\frac{{\sqrt {s - u} }}{2}({\eta _1} - {\eta _0}), c),$

(3.30)

where ${c^2} = \frac{{v - u}}{{s - u}}.$

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_8}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}}[u + (v - u) \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)]} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.31)

${q_8}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{3}}}}[u + (v - u) \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)].$

(3.32)

When $\omega \gt s,$ the solution of Eq (3.7) is

${\omega _9} = \frac{{ - v \cdot s{n^2}(\sqrt {s - u} ({\eta _1} - {\eta _0})/2, c) + s}}{{c{n^2}(\sqrt {s - u} ({\eta _1} - {\eta _0})/2, c)}}.$

(3.33)

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_9}(x, y, t) = \pm \sqrt { - {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}}\frac{{ - v \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)] + s}}{{c{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}}} {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}}.$

(3.34)

${q_9}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{3}}}}\frac{{ - v \cdot s{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)] + s}}{{c{n^2}(\frac{{\sqrt {s - u} }}{2}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}}.$

(3.35)

Case 4: $\Delta \lt 0.$

When $\omega \gt 0,$ similarly we get

${\omega _{10}} = \frac{{2\sqrt {{b_0}} }}{{1 + cn(b_0^{\frac{1}{4}}({\eta _1} - {\eta _0}), c)}} - \sqrt {{b_0}} ,$

(3.36)

where ${c^2} = (1 - \frac{{{b_1}\sqrt {{b_0}} }}{2})/2.$

The exact traveling wave solutions of the Eq (1.1) can be expressed as below:

${p_{10}}(x, y, t) = \pm \sqrt {2\sqrt {{a_0}} {{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{2}}}}[\frac{{ - 2}}{{1 + cn({{(4{a_0}{{( - \frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}})}^{\frac{1}{4}}}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}} + 1]} \cdot {e^{i({\lambda _1}x + {\lambda _2}y + {\lambda _3}t + {\lambda _4})}} ,$

(3.37)

${q_{10}}(x, y, t) = - \frac{{{r_4}}}{{{r_3}}}2\sqrt {{a_0}} {(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})^{^{ - \frac{1}{2}}}}[\frac{{ - 2}}{{1 + cn({{(4{a_0}{{( - \frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{ - \frac{1}{3}}}})}^{\frac{1}{4}}}({{(\frac{{2{r_2}{r_4}}}{{{r_1}{r_3}}})}^{^{\frac{1}{3}}}}\eta + {\eta _0}), c)}} + 1],$

(3.38)

where ${p_8}(x, y, t), {q_8}(x, y, t), {p_9}(x, y, t), {q_9}(x, y, t), {p_{10}}(x, y, t)$ and ${q_{10}}(x, y, t)$ are Jacobian elliptic function solutions.

4. Conclusions

In this paper, the complete discrimination system of polynomial method has been applied to construct the single traveling wave solutions of the Fokas system. The Jacobian elliptic function solutions, the trigonometric function solutions, the hyperbolic function solutions and the rational function solutions are obtained. The obtained solutions are very rich, which can help physicists understand the propagation of traveling wave in monomode optical fibers. Furthermore, we have also depicted two-dimensional graphs, three-dimensional graphs, contour plots and density plots of the solutions of Fokas system, which explains the state of solitons from different angles.

Acknowledgments

This work was supported by Scientific Research Funds of Chengdu University (Grant No.2081920034).

Conflict of interest

The authors declare no conflict of interest.

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Mathematical Biosciences and Engineering

Synchronization analysis of coupled fractional-order neural networks with time-varying delays

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1. Introduction

2. Description of the method

3. Traveling wave solution of Eq (1.1)

4. Conclusions

Acknowledgments

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Mathematical Biosciences and Engineering

Synchronization analysis of coupled fractional-order neural networks with time-varying delays

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1. Introduction

2. Description of the method

3. Traveling wave solution of Eq (1.1)

4. Conclusions

Acknowledgments

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