Complexiton solutions and periodic-soliton solutions for the (2+1)-dimensional BLMP equation

Jian-Guo Liu; Wen-Hui Zhu; Yan He; Aly R. Seadawy; Jian-Guo Liu; Wen-Hui Zhu; Yan He; Aly R. Seadawy

doi:10.3934/math.2020029

AIMS Mathematics

2020, Volume 5, Issue 1: 421-439. doi: 10.3934/math.2020029

Previous Article Next Article

Research article

Complexiton solutions and periodic-soliton solutions for the (2+1)-dimensional BLMP equation

Jian-Guo Liu ^{1
,
,},
Wen-Hui Zhu ^{2
,
,},
Yan He ^{1
,
,},
Aly R. Seadawy ^{3
,
,}

1.
College of Computer, Jiangxi University of Traditional Chinese Medicine, Jiangxi 330004, China
2.
Institute of artificial intelligence, Nanchang Institute of Science and Technology, Jiangxi 330108, China
3.
Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

Received: 29 October 2019 Accepted: 02 December 2019 Published: 04 December 2019
MSC : 35C08, 68M07, 33F10

The (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is studied, which describes the incompressible fluid. By virtue of an ansätz functions and the bilinear form, many entirely new complexiton solutions and periodic-soliton solutions are derived. With the aid of symbolic computation, their dynamical behaviors are demonstrated in some three-dimensional plots by choosing different values of the parameters.

Keywords:

Citation: Jian-Guo Liu, Wen-Hui Zhu, Yan He, Aly R. Seadawy. Complexiton solutions and periodic-soliton solutions for the (2+1)-dimensional BLMP equation[J]. AIMS Mathematics, 2020, 5(1): 421-439. doi: 10.3934/math.2020029

Related Papers:

[1]	Sixing Tao . Breathers, resonant multiple waves and complexiton solutions of a (2+1)-dimensional nonlinear evolution equation. AIMS Mathematics, 2023, 8(5): 11651-11665. doi: 10.3934/math.2023590
[2]	Jianhong Zhuang, Yaqing Liu, Ping Zhuang . Variety interaction solutions comprising lump solitons for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation. AIMS Mathematics, 2021, 6(5): 5370-5386. doi: 10.3934/math.2021316
[3]	M. A. El-Shorbagy, Sonia Akram, Mati ur Rahman, Hossam A. Nabwey . Analysis of bifurcation, chaotic structures, lump and $ M-W $-shape soliton solutions to $ (2+1) $ complex modified Korteweg-de-Vries system. AIMS Mathematics, 2024, 9(6): 16116-16145. doi: 10.3934/math.2024780
[4]	Gaukhar Shaikhova, Bayan Kutum, Ratbay Myrzakulov . Periodic traveling wave, bright and dark soliton solutions of the (2+1)-dimensional complex modified Korteweg-de Vries system of equations by using three different methods. AIMS Mathematics, 2022, 7(10): 18948-18970. doi: 10.3934/math.20221043
[5]	Ninghe Yang . Exact wave patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional NLSE. AIMS Mathematics, 2024, 9(11): 31274-31294. doi: 10.3934/math.20241508
[6]	Naveed Iqbal, Meshari Alesemi . Soliton dynamics in the $ (2+1) $-dimensional Nizhnik-Novikov-Veselov system via the Riccati modified extended simple equation method. AIMS Mathematics, 2025, 10(2): 3306-3333. doi: 10.3934/math.2025154
[7]	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
[8]	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
[9]	Ahmed A. Gaber, Abdul-Majid Wazwaz . Dynamic wave solutions for (2+1)-dimensional DJKM equation in plasma physics. AIMS Mathematics, 2024, 9(3): 6060-6072. doi: 10.3934/math.2024296
[10]	Noor Alam, Mohammad Safi Ullah, Jalil Manafian, Khaled H. Mahmoud, A. SA. Alsubaie, Hamdy M. Ahmed, Karim K. Ahmed, Soliman Al Khatib . Bifurcation analysis, chaotic behaviors, and explicit solutions for a fractional two-mode Nizhnik-Novikov-Veselov equation in mathematical physics. AIMS Mathematics, 2025, 10(3): 4558-4578. doi: 10.3934/math.2025211

Abstract

1. Introduction

The nonlinear evolution equations (NLEEs) can be used to describe many physical models ^[1,2,3,4,5]. The investigation on exact solutions and numerical solutions of NLEEs has become one of the most important areas in the study of nonlinear physical phenomena ^[6,7,8,9]. Via symbolic computation ^{[10,11,12,13,14,15,16,17]}, many effective methods are presented ^{[18,19,20,21,22,23,24,25]}.

In this paper, a (2+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation is investigated as ^{[26,27,28,29,30,31,32,33,34,35]}

$\begin{eqnarray} u_{yt}+u_{xxxy}-3\,u_x\,u_{xy}-3\,u_{xx}\,u_y = 0, \end{eqnarray}$

(1.1)

where $u = u(x, y, t)$ . Some exact solutions including kinky periodic solitary-wave solutions, periodic soliton solutions and kink solutions were discussed ^[26]. Bilinear form was presented via using the binary Bell polynomials ^[27]. The variable separable solutions and some novel localized excitations were obtained ^[28]. New solutions were derived via wronskian formalism and the Hirota method ^[29,30]. The periodic-soliton solutions are investigated ^[31] and so on ^{[32,33,34,35]}. But so far, complexiton solutions and double periodic-soliton solutions for Eq. (1) have not been obtained.

The organization of this paper is as follows. Section 2 obtains many new complexiton solutions and double periodic-soliton solutions based the bilinear form and an ansätz function, their dynamical behaviors are demonstrated in some three-dimensional plots by selecting different values of the parameters. Section 3 gives the conclusions.

2. Complexiton solutions and double periodic-soliton solutions

Under the transformation

$\begin{eqnarray} u(x,y,t) = -2\,[\ln \Psi(x,y,t)]_x, \end{eqnarray}$

(2.1)

the Eq. (1) is transformed into the bilinear form

$\begin{eqnarray} (D_y D_t+D_y D_x^3) \Psi \cdot \Psi = 0. \end{eqnarray}$

(2.2)

Eq. (3) is equivalent to

$\begin{eqnarray} -\Psi_t \Psi_y-\Psi_{xxx} \Psi_y+3 \Psi_{xy} \Psi_{xx}-3 \Psi_x \Psi_{xxy}+\Psi \left(\Psi_{yt}+\Psi_{xxxy}\right) = 0. \end{eqnarray}$

(2.3)

Supposing Eq. (4) has the following form of solution:

$\begin{eqnarray} \Psi& = & e^{\Psi _1} [\Theta_1 \cos(\Psi _2)+\Theta_2 \sin(\Psi _2)]+k_1 e^{2 \Psi _1}\\&+&e^{\Psi _3} [\Theta_3 \cos(\Psi _4)+\Theta_4 \sin(\Psi _4)]+k_2 e^{\Psi _4}, \end{eqnarray}$

(2.4)

where $\Psi_i = \iota_i\, x+\rho_i\, y+\varsigma_i\, t, i = 1, 2, 3, 4$ and $\iota _i$ , $\rho _i$ and $\varsigma _i$ are unknown constants. Substituting Eq. (5) into Eq. (4) and equating corresponding coefficients of $e^{\Psi _1}$ , $e^{\Psi _3}$ , $e^{\Psi _4}$ , $\cos \Psi _2$ , $\sin \Psi _2$ , $\cos \Psi _4$ , and $\sin \Psi _4$ to zero, a set of algebraic equations for $\iota _i$ , $\rho _i$ and $\varsigma _i$ can be presented as follows

Case (1)

$\begin{eqnarray} k_1 & = & \rho_4 = \rho_2 = 0, \rho_1 = \rho_3, \varsigma_4 = \iota _4 \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right),\\ \varsigma_2& = &\iota _2 \left(-3 \iota _1^2+12 \iota _4 \iota _1+\iota _2^2-12 \iota _4^2\right), \varsigma_3 = -\iota _3^3+15 \iota _4^2 \iota _3-20 \iota _4^3,\\ \varsigma_1& = &-\iota _1^3+6 \iota _4 \iota _1^2+3 \iota _2^2 \iota _1-12 \iota _4^2 \iota _1-14 \iota _4^3 \\&+&24 \iota _3 \iota _4^2-6 \iota _2^2 \iota _4-6 \iota _3^2 \iota _4, \end{eqnarray}$

(2.5)

$\begin{eqnarray} \Psi & = & e^{2 \left(x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4\right)} k_2+e^{x \iota _1+y \rho _3+t \varsigma _1} [\cos [x \iota _2+t (-3 \iota _1^2\\&+&12 \iota _4 \iota _1+\iota _2^2-12 \iota _4^2) \iota _2] \Theta _1 +\sin [x \iota _2+t (-3 \iota _1^2+12 \iota _4 \iota _1+\iota _2^2\\&-&12 \iota _4^2) \iota _2] \Theta _2]+e^{x \iota _3+t \left(-\iota _3^3+15 \iota _4^2 \iota _3-20 \iota _4^3\right)+y \rho _3} [\cos [x \iota _4+t (-3 \iota _3^2\\&+&12 \iota _4 \iota _3-11 \iota _4^2) \iota _4] \Theta _3+\sin [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _4]. \end{eqnarray}$

(2.6)

$\begin{eqnarray} u_1 & = & -[2 [2 e^{2 [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2 \iota _4+e^{x \iota _1+y \rho _3+t \varsigma _1} \iota _1 [\cos [x \iota _2\\&+&t \left(-3 \iota _1^2+12 \iota _4 \iota _1 +\iota _2^2-12 \iota _4^2\right) \iota _2] \Theta _1+\sin [x \iota _2+t (-3 \iota _1^2+12 \iota _4 \iota _1\\&+&\iota _2^2-12 \iota _4^2) \iota _2] \Theta _2]+e^{x \iota _1+y \rho _3+t \varsigma _1} [\cos [x \iota _2+t (-3 \iota _1^2+12 \iota _4 \iota _1 +\iota _2^2\\&-&12 \iota _4^2) \iota _2] \iota _2 \Theta _2-\sin [x \iota _2+t \left(-3 \iota _1^2+12 \iota _4 \iota _1+\iota _2^2 -12 \iota _4^2\right) \iota _2] \iota _2 \Theta _1]\\&+&e^{x \iota _3+t \left(-\iota _3^3+15 \iota _4^2 \iota _3-20 \iota _4^3\right)+y \rho _3} \iota _3 [\cos [x \iota _4+t (-3 \iota _3^2+12 \iota _4 \iota _3\\&-&11 \iota _4^2) \iota _4] \Theta _3+\sin \left(x \iota _4 +t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4\right) \Theta _4]\\&+&e^{x \iota _3+t \left(-\iota _3^3+15 \iota _4^2 \iota _3 -20 \iota _4^3\right)+y \rho _3} [\cos [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]\\&*& \iota _4 \Theta _4 -\sin [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \iota _4 \Theta _3]]]\\&/&[e^{2 [x \iota _4 +t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2+e^{x \iota _1+y \rho _3+t \varsigma _1} [\cos [x \iota _2 +t (-3 \iota _1^2\\&+&12 \iota _4 \iota _1+\iota _2^2-12 \iota _4^2) \iota _2] \Theta _1+\sin [x \iota _2+t (-3 \iota _1^2 +12 \iota _4 \iota _1+\iota _2^2\\&-&12 \iota _4^2) \iota _2] \Theta _2]+e^{x \iota _3+t \left(-\iota _3^3+15 \iota _4^2 \iota _3 -20 \iota _4^3\right)+y \rho _3} [\cos [x \iota _4+t (-3 \iota _3^2+12 \iota _4\\&*& \iota _3-11 \iota _4^2) \iota _4] \Theta _3 +\sin [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _4]]. \end{eqnarray}$

(2.7)

The dynamical behavior to Eq. (8) is demonstrated in Figure 1.

Figure 1.

$\iota_1 = k_2 = \iota_3 = \Theta_2 = -1$ ,

$\iota_2 = \iota_4 = 0$ ,

$\Theta_1 = \rho_3 = \Theta_3 = \Theta_4 = 1$ , (a)

$t = -10$ , (b)

$t = 0$ and (c)

$t = 10$ .

DownLoad: Full-Size Img PowerPoint

Case (2)

$\begin{eqnarray} k_1 & = & \rho_4 = \iota_2 = 0, \varsigma_2 = 4 \iota _2^3, \iota_1 = 2 \iota_4, \varsigma_4 = \iota _4 \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right), \\ \varsigma_1& = &-22 \iota _4^3+24 \iota _3 \iota _4^2-6 \iota _3^2 \iota _4, \varsigma_3 = -\iota _3^3+15 \iota _4^2 \iota _3-20 \iota _4^3, \end{eqnarray}$

(2.8)

$\begin{eqnarray} \Psi & = & e^{2 [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2+e^{2 x \iota _4+t \left(-22 \iota _4^3+24 \iota _3 \iota _4^2-6 \iota _3^2 \iota _4\right)+y \rho _1} [\cos \\&&\left(y \rho _2\right) \Theta _1 +\sin \left(y \rho _2\right) \Theta _2]+e^{x \iota _3+t \left(-\iota _3^3+15 \iota _4^2 \iota _3-20 \iota _4^3\right)+y \rho _3} [\cos [x \iota _4 +t (-3 \iota _3^2\\&+&12 \iota _4 \iota _3-11 \iota _4^2) \iota _4] \Theta _3+\sin [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3 -11 \iota _4^2\right) \iota _4] \Theta _4]. \end{eqnarray}$

(2.9)

$\begin{eqnarray} u_2 & = & -[2 [2 e^{2 [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2 \iota _4+2 [\cos \left(y \rho _2\right) \Theta _1 +\sin [y \rho _2] \Theta _2] \iota _4\\&& e^{2 x \iota _4+t \left(-22 \iota _4^3+24 \iota _3 \iota _4^2-6 \iota _3^2 \iota _4\right)+y \rho _1} +e^{x \iota _3+t \left(-\iota _3^3 +15 \iota _4^2 \iota _3-20 \iota _4^3\right)+y \rho _3} \iota _3 \\&&[\cos [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _3+\sin [x \iota _4 +t (-3 \iota _3^2\\&+&12 \iota _4 \iota _3-11 \iota _4^2) \iota _4] \Theta _4]+e^{x \iota _3+t \left(-\iota _3^3+15 \iota _4^2 \iota _3 -20 \iota _4^3\right)+y \rho _3} [\cos [x \iota _4\\&+&t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \iota _4 \Theta _4 -\sin [x \iota _4+t (-3 \iota _3^2+12 \iota _4 \iota _3\\&-&11 \iota _4^2) \iota _4] \iota _4 \Theta _3]]]/[e^{2 [x \iota _4 +t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2+[\cos \left(y \rho _2\right) \Theta _1\\&+& \sin \left(y \rho _2\right) \Theta _2] e^{2 x \iota _4+t \left(-22 \iota _4^3+24 \iota _3 \iota _4^2 -6 \iota _3^2 \iota _4\right)+y \rho _1} + [\cos [x \iota _4+t (-3 \iota _3^2\\&+&12 \iota _4 \iota _3 -11 \iota _4^2) \iota _4] \Theta _3+\sin [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _4]]\\&*& e^{x \iota _3 +t \left(-\iota _3^3+15 \iota _4^2 \iota _3-20 \iota _4^3\right)+y \rho _3}. \end{eqnarray}$

(2.10)

The dynamical behavior to Eq. (11) is demonstrated in Figure 2.

Figure 2.

$\Theta_3 = \Theta_2 = -1$ ,

$k_2 = 2$ ,

$\Theta_1 = \iota_3 = \iota_4 = \rho_1 = \rho_2 = \rho_3 = \Theta_4 = 1$ , (a)

$t = -2$ , (b)

$t = 0$ and (c)

$t = 2$ .

DownLoad: Full-Size Img PowerPoint

Case (3)

$\begin{eqnarray} k_1 & = & \rho_2 = \iota_4 = 0, \varsigma_2 = \iota _2 \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right), \rho_1 = 2 \rho_4, \varsigma_4 = 4 \iota _4^3, \\ \varsigma_1& = &[-\Theta _2 \iota _1^3+3 \left(\iota _2 \Theta _1+\iota _3 \Theta _2\right) \iota _1^2 +3 \left(\left(\iota _2^2-\iota _3^2\right) \Theta _2-2 \iota _2 \iota _3 \Theta _1\right) \iota _1-\iota _2^3 \Theta _1\\&+&3 \iota _2 \iota _3^2 \Theta _1-3 \iota _2^2 \iota _3 \Theta _2 +\Theta _1 \varsigma _2]/\Theta _2, \varsigma_3 = -\iota _3^3, \end{eqnarray}$

(2.11)

$\begin{eqnarray} \Psi & = & e^{2 y \rho _4} k_2+e^{x \iota _1+2 y \rho _4+t \varsigma _1} [\cos [x \iota _2+t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \Theta _1\\&+&\sin [x \iota _2+t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \Theta _2]\\&+&e^{-t \iota _3^3+x \iota _3+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.12)

$\begin{eqnarray} u_3 & = & -[2 [e^{x \iota _1+2 y \rho _4+t \varsigma _1} \iota _1 [\cos [x \iota _2+t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \Theta _1\\&+&\sin [x \iota _2+t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \Theta _2]+e^{x \iota _1+2 y \rho _4+t \varsigma _1} [\cos [x \iota _2\\&+&t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \iota _2 \Theta _2-\sin [x \iota _2+t (-3 \iota _1^2+6 \iota _3 \iota _1\\&+&\iota _2^2-3 \iota _3^2) \iota _2] \iota _2 \Theta _1]+e^{-t \iota _3^3+x \iota _3+y \rho _3} \iota _3 [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]]\\&/&[e^{2 y \rho _4} k_2+e^{x \iota _1+2 y \rho _4+t \varsigma _1} [\cos [x \iota _2+t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \Theta _1\\&+&\sin [x \iota _2+t \left(-3 \iota _1^2+6 \iota _3 \iota _1+\iota _2^2-3 \iota _3^2\right) \iota _2] \Theta _2]\\&+&e^{-t \iota _3^3+x \iota _3+y \rho _3} \left(\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4\right)]. \end{eqnarray}$

(2.13)

The dynamical behavior to Eq. (14) is demonstrated in Figure 3.

Figure 3.

$\Theta_3 = \Theta_2 = -1$ ,

$k_2 = 2$ ,

$\Theta_1 = \iota_3 = \iota_2 = \iota_1 = \rho_4 = \rho_3 = \Theta_4 = 1$ , (a)

$t = -5$ , (b)

$t = 0$ and (c)

$t = 5$ .

DownLoad: Full-Size Img PowerPoint

Case (4)

$\begin{eqnarray} k_1 & = & \rho_2 = \varsigma_3 = \iota_3 = \iota_4 = \varsigma_4 = 0,\\ \varsigma_1& = &3 \iota _1 \iota _2^2-\iota _1^3, \varsigma_2 = \iota _2 \left(\iota _2^2-3 \iota _1^2\right), \end{eqnarray}$

(2.14)

$\begin{eqnarray} \Psi & = & e^{2 y \rho _4} k_2+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _1+\sin [x \iota _2\\&+&t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _2]+e^{y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.15)

$\begin{eqnarray} u_4 & = & -[2 [e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} \iota _1 [\cos \left(x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2\right) \Theta _1+\sin [x \iota _2\\&+&t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _2]+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2]\\&*& \iota _2 \Theta _2-\sin [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \iota _2 \Theta _1]]]/[[\cos [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _1\\&+&\sin \left(x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2\right) \Theta _2] e^{2 y \rho _4} k_2+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1}\\&+&e^{y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.16)

The dynamical behavior to Eq. (17) is demonstrated in Figure 4.

Figure 4.

$\iota_1 = \Theta_2 = -1$ ,

$k_2 = -2$ ,

$\Theta_1 = \rho_1 = \rho_3 = \rho_4 = \Theta_3 = \Theta_4 = 1$ ,

$\iota_2 = 0$ , (a)

$x = -10$ , (b)

$x = 0$ and (c)

$x = 10$ .

DownLoad: Full-Size Img PowerPoint

Case (5)

$\begin{eqnarray} k_1 = \varsigma_2 = \iota_2 = \iota_4 = \varsigma_4 = 0, \iota_3 = \iota_1, \varsigma_3 = -\iota _1^3, \varsigma_1 = - \iota _1^3, \end{eqnarray}$

(2.17)

$\begin{eqnarray} \Psi & = & e^{2 y \rho _4} k_2+e^{-t \iota _1^3+x \iota _1+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]\\&+&e^{-t \iota _1^3+x \iota _1+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.18)

$\begin{eqnarray} u_5 & = & -[2 [e^{-t \iota _1^3+x \iota _1+y \rho _1} \iota _1 [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]+e^{-t \iota _1^3+x \iota _1+y \rho _3}\\&*& \iota _1 [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]]/[e^{2 y \rho _4} k_2+e^{-t \iota _1^3+x \iota _1+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2]+e^{-t \iota _1^3+x \iota _1+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.19)

The dynamical behavior to Eq. (20) is demonstrated in Figure 5.

Figure 5.

$\iota_1 = \Theta_2 = -1$ ,

$k_2 = -2$ ,

$\Theta_1 = \rho_1 = \rho_2 = \rho_3 = \rho_4 = \Theta_3 = \Theta_4 = 1$ , (a)

$x = -10$ , (b)

$x = 0$ and (c)

$x = 10$ .

DownLoad: Full-Size Img PowerPoint

Case (6)

$\begin{eqnarray} k_2 & = & \varsigma_2 = \rho_4 = \iota_2 = 0, \varsigma_3 = -\iota _3^3+3 \iota _1 \iota _3^2-3 \iota _1^2 \iota _3+3 \iota _4^2 \iota _3-3 \iota _1 \iota _4^2,\\ \varsigma_4& = &\iota _4^3-3 \left(\iota _1-\iota _3\right)^2 \iota _4, \varsigma_1 = -\iota _1^3, \rho_3 = 2 \rho_1, \end{eqnarray}$

(2.20)

$\begin{eqnarray} \Psi & = & e^{2 \left(-t \iota _1^3+x \iota _1+y \rho _1\right)} k_1+e^{-t \iota _1^3+x \iota _1+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]\\&+&e^{x \iota _3+t \left(-\iota _3^3+3 \iota _1 \iota _3^2-3 \iota _1^2 \iota _3+3 \iota _4^2 \iota _3-3 \iota _1 \iota _4^2\right)+2 y \rho _1} [\cos [x \iota _4+t \varsigma_4] \Theta _3\\&+&\sin [x \iota _4+t \left(\iota _4^3-3 \left(\iota _1-\iota _3\right)^2 \iota _4\right)] \Theta _4]. \end{eqnarray}$

(2.21)

$\begin{eqnarray} u_6 & = & -[2 [2 e^{2 \left(-t \iota _1^3+x \iota _1+y \rho _1\right)} k_1 \iota _1+e^{-t \iota _1^3+x \iota _1+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right)\\&*& \Theta _2] \iota _1+e^{x \iota _3+2 y \rho _1+t \varsigma _3} \iota _3 \left(\sin \left(x \iota _4+t \varsigma _4\right)+\cos \left(x \iota _4+t \varsigma _4\right) \Theta _3\right)\\&+&e^{x \iota _3+2 y \rho _1+t \varsigma _3} [\cos \left(x \iota _4+t \varsigma _4\right) \iota _4-\sin \left(x \iota _4+t \varsigma _4\right) \iota _4 \Theta _3]]]\\&/&[e^{2 \left(-t \iota _1^3 +x \iota _1+y \rho _1\right)} k_1+e^{-t \iota _1^3+x \iota _1+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]\\&+&e^{x \iota _3+2 y \rho _1+t \varsigma _3} [\sin \left(x \iota _4+t \varsigma _4\right)+\cos \left(x \iota _4+t \varsigma _4\right) \Theta _3]]. \end{eqnarray}$

(2.22)

The dynamical behavior to Eq. (23) is demonstrated in Figure 6.

Figure 6.

$\iota_3 = \Theta_2 = \Theta_1 = -1$ ,

$k_1 = 2$ ,

$\iota_1 = \rho_1 = \rho_2 = \Theta_3 = \Theta_4 = 1$ , (a)

$y = -5$ , (b)

$y = 0$ and (c)

$y = 5$ .

DownLoad: Full-Size Img PowerPoint

Case (7)

$\begin{eqnarray} k_2 = \varsigma_2 = \varsigma_1 = \rho_4 = \iota_2 = \iota_1 = 0, \varsigma_3 = 3 \iota _3 \iota _4^2-\iota _3^3, \varsigma_4 = \iota _4^3-3 \iota _3^2 \iota _4, \end{eqnarray}$

(2.23)

$\begin{eqnarray} \Psi & = & e^{2 y \rho _1} k_1+e^{y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]+e^{x \iota _3+t \left(3 \iota _3 \iota _4^2-\iota _3^3\right)+y \rho _3}\\&*& [\cos [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \Theta _3+\sin [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \Theta _4]. \end{eqnarray}$

(2.24)

$\begin{eqnarray} u_7 & = & -[2 [e^{x \iota _3+t \left(3 \iota _3 \iota _4^2-\iota _3^3\right)+y \rho _3} \iota _3 [\cos [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \Theta _3+\sin [x \iota _4\\&+&t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \Theta _4] +e^{x \iota _3+t \left(3 \iota _3 \iota _4^2-\iota _3^3\right)+y \rho _3} [\cos [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)]\\&*& \iota _4 \Theta _4-\sin [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \iota _4 \Theta _3]]]/[e^{2 y \rho _1} k_1 +e^{y \rho _1} [\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2]+e^{x \iota _3 +t \left(3 \iota _3 \iota _4^2-\iota _3^3\right)+y \rho _3} [\cos [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \Theta _3 \\&+&\sin [x \iota _4+t \left(\iota _4^3-3 \iota _3^2 \iota _4\right)] \Theta _4]]. \end{eqnarray}$

(2.25)

The dynamical behavior to Eq. (26) is demonstrated in Figure 7.

Figure 7.

$\iota_3 = \Theta_2 = \Theta_1 = -1$ ,

$k_1 = 2$ ,

$\iota_4 = 0$ ,

$\rho_1 = \rho_2 = \rho_3 = \Theta_3 = \Theta_4 = 1$ , (a)

$t = -5$ , (b)

$t = 0$ and (c)

$t = 5$ .

DownLoad: Full-Size Img PowerPoint

Case (8)

$\begin{eqnarray} k_2& = &\rho_4 = \rho_2 = 0, \rho_1 = \rho_3, \varsigma_1 = 3 \iota _1 \iota _2^2-\iota _1^3, \varsigma_2 = \iota _2^3-3 \iota _1^2 \iota _2,\\ \varsigma_3& = &6 \iota _1^3-12 \iota _3 \iota _1^2+6 \iota _2^2 \iota _1+6 \iota _3^2 \iota _1-6 \iota _4^2 \iota _1-\iota _3^3+3 \iota _3 \iota _4^2,\\ \varsigma_4& = &\iota _4^3-3 \left(\iota _3-2 \iota _1\right)^2 \iota _4, \end{eqnarray}$

(2.26)

$\begin{eqnarray} \Psi & = & e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3]} k_1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3} [\cos [x \iota _2+t (\iota _2^3\\&-&3 \iota _1^2 \iota _2)] \Theta _1+\sin [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _2]+e^{x \iota _3+y \rho _3+t \varsigma _3}\\&*& [\sin \left(x \iota _4+t \varsigma _4\right)+\cos \left(x \iota _4+t \varsigma _4\right) \Theta _3]. \end{eqnarray}$

(2.27)

$\begin{eqnarray} u_8 & = & -[2 [2 e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3]} k_1 \iota _1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3} [\cos [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _1+\sin [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _2] \iota _1\\&+&e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3} [\cos [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \iota _2 \Theta _2-\sin [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \iota _2 \Theta _1]+e^{x \iota _3+y \rho _3+t \varsigma _3} \iota _3 [\sin \left(x \iota _4+t \varsigma _4\right)+\cos (x \iota _4\\&+&t \varsigma _4) \Theta _3]+e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos \left(x \iota _4+t \varsigma _4\right) \iota _4-\sin \left(x \iota _4+t \varsigma _4\right) \iota _4 \Theta _3]]]\\&/&[e^{2 \left(x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3\right)} k_1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _3} [\cos [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _1+\sin \left(x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)\right) \Theta _2]\\&+&e^{x \iota _3+y \rho _3+t \varsigma _3} \left(\sin \left(x \iota _4+t \varsigma _4\right)+\cos \left(x \iota _4+t \varsigma _4\right) \Theta _3\right)]. \end{eqnarray}$

(2.28)

The dynamical behavior to Eq. (29) is demonstrated in Figure 8.

Figure 8.

$\iota_1 = \rho_3 = \Theta_3 = \Theta_2 = 1$ ,

$\Theta_4 = 1$ ,

$\iota_3 = \iota_4 = \iota_2 = k_1 = \Theta_1 = -1$ , (a)

$y = -15$ , (b)

$y = 0$ and (c)

$y = 15$ .

DownLoad: Full-Size Img PowerPoint

Case (9)

$\begin{eqnarray} k_2& = &\varsigma_4 = \varsigma_2 = \iota_2 = \iota_4 = 0, \iota_1 = \iota_3, \varsigma_1 = -\iota _3^3, \varsigma_3 = -\iota _3^3, \end{eqnarray}$

(2.29)

$\begin{eqnarray} \Psi & = & e^{2 \left(-t \iota _3^3+x \iota _3+y \rho _1\right)} k_1+e^{-t \iota _3^3+x \iota _3+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]\\&+&e^{-t \iota _3^3+x \iota _3+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.30)

$\begin{eqnarray} u_9 & = & -[2 [2 e^{2 \left(-t \iota _3^3+x \iota _3+y \rho _1\right)} k_1 \iota _3+e^{-t \iota _3^3+x \iota _3+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]\\&*& \iota _3+e^{-t \iota _3^3+x \iota _3+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4] \iota _3]]/[e^{2 \left(-t \iota _3^3+x \iota _3+y \rho _1\right)} k_1 \\&+&e^{-t \iota _3^3+x \iota _3+y \rho _1} [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]\\&+&e^{-t \iota _3^3+x \iota _3 +y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.31)

The dynamical behavior to Eq. (32) is demonstrated in Figure 9.

Figure 9.

$\rho_1 = \rho_4 = \rho_2 = k_1 = 1$ ,

$\rho_3 = \Theta_3 = \Theta_2 = \Theta_4 = 1$ ,

$\iota_3 = \Theta_1 = -1$ , (a)

$t = -5$ , (b)

$t = 0$ and (c)

$t = 5$ .

DownLoad: Full-Size Img PowerPoint

Case (10)

$\begin{eqnarray} k_2& = &\varsigma_4 = \rho_2 = \iota_4 = 0, \iota_3 = 2 \iota_1, \varsigma_1 = 3 \iota _1 \iota _2^2-\iota _1^3,\\ \varsigma_2& = &\iota _2^3-3 \iota _1^2 \iota _2, \varsigma_3 = 6 \iota _1 \left(4 \iota _1^2+\iota _2^2\right)-26 \iota _1^3, \end{eqnarray}$

(2.32)

$\begin{eqnarray} \Psi & = & e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1]} k_1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2+t (\iota _2^3\\&-&3 \iota _1^2 \iota _2)] \Theta _1 +\sin [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _2]\\&+&e^{2 x \iota _1 +t [6 \iota _1 \left(4 \iota _1^2+\iota _2^2\right)-26 \iota _1^3]+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.33)

$\begin{eqnarray} u_{10} & = & -[2 [2 e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1]} k_1 \iota _1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _1+\sin [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _2] \iota _1+2 e^{2 x \iota _1+y \rho _3+t \varsigma _3} \\&*&[\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4] \iota _1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2 -\iota _1^3\right)+y \rho _1} [\cos [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \iota _2 \Theta _2-\sin [x \iota _2 +t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \iota _2 \Theta _1]]]\\&/&[e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2 -\iota _1^3\right)+y \rho _1]} k_1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2\\&+&t \left(\iota _2^3 -3 \iota _1^2 \iota _2\right)] \Theta _1+\sin [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _2] \\&+&e^{2 x \iota _1+y \rho _3+t \varsigma _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.34)

The dynamical behavior to Eq. (35) is demonstrated in Figure 10.

Figure 10.

$\rho_1 = \Theta_4 = k_1 = 1$ ,

$\iota_1 = 0$ ,

$\rho_4 = \Theta_3 = \Theta_2 = \Theta_1 = 1$ ,

$\iota_2 = -2$ , (a)

$t = -25$ , (b)

$t = 0$ and (c)

$t = 25$ .

DownLoad: Full-Size Img PowerPoint

Case (11)

$\begin{eqnarray} \rho_1& = &\rho_4 = \varsigma_2 = \iota_2 = \rho_3 = 0, \iota_1 = \frac{3 \iota _3^2-5 \iota _4^2}{6 \left(\iota _3-\iota _4\right)}, \varsigma_1 = -\frac{\left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3},\\ \varsigma_3& = &\frac{-3 \iota _3^5+6 \iota _4 \iota _3^4+6 \iota _4^2 \iota _3^3-24 \iota _4^3 \iota _3^2+41 \iota _4^4 \iota _3-30 \iota _4^5}{12 \left(\iota _3-\iota _4\right)^2},\\ \varsigma_4& = &-\frac{\iota _4 \left(9 \iota _3^4-36 \iota _4 \iota _3^3+54 \iota _4^2 \iota _3^2-36 \iota _4^3 \iota _3+13 \iota _4^4\right)}{12 \left(\iota _3-\iota _4\right)^2}, \end{eqnarray}$

(2.35)

$\begin{eqnarray} \Psi & = & e^{2 [\frac{x \left(3 \iota _3^2-5 \iota _4^2\right)}{6 \left(\iota _3-\iota _4\right)}-\frac{t \left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3}]} k_1+e^{2 \left(x \iota _4+t \varsigma _4\right)} k_2+e^{\frac{x \left(3 \iota _3^2-5 \iota _4^2\right)}{6 \left(\iota _3-\iota _4\right)}-\frac{t \left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3}}\\&*& [\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2]+e^{x \iota _3+t \varsigma _3} [\cos \left(x \iota _4 +t \varsigma _4\right) \Theta _3\\&+&\sin \left(x \iota _4+t \varsigma _4\right) \Theta _4]. \end{eqnarray}$

(2.36)

$\begin{eqnarray} u_{11} & = & -[2 [2 e^{2 \left(x \iota _4+t \varsigma _4\right)} k_2 \iota _4+\frac{e^{2 [\frac{x \left(3 \iota _3^2-5 \iota _4^2\right)}{6 \left(\iota _3-\iota _4\right)}-\frac{t \left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3}]} k_1 \left(3 \iota _3^2-5 \iota _4^2\right)}{3 \left(\iota _3-\iota _4\right)}\\&+&\frac{e^{\frac{x \left(3 \iota _3^2-5 \iota _4^2\right)}{6 \left(\iota _3 -\iota _4\right)}-\frac{t \left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3}} \left(3 \iota _3^2-5 \iota _4^2\right) \left(\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2\right)}{6 \left(\iota _3-\iota _4\right)}\\&+&e^{x \iota _3+t \varsigma _3} \iota _3 \left(\cos \left(x \iota _4+t \varsigma _4\right) \Theta _3+\sin \left(x \iota _4+t \varsigma _4\right) \Theta _4\right)\\&+&e^{x \iota _3+t \varsigma _3} [\cos \left(x \iota _4+t \varsigma _4\right) \iota _4 \Theta _4-\sin \left(x \iota _4+t \varsigma _4\right) \iota _4 \Theta _3]]]\\&/&[e^{2 \left(\frac{x \left(3 \iota _3^2 -5 \iota _4^2\right)}{6 \left(\iota _3-\iota _4\right)}-\frac{t \left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3}\right)} k_1 +e^{2 \left(x \iota _4+t \varsigma _4\right)} k_2\\&+&e^{\frac{x \left(3 \iota _3^2-5 \iota _4^2\right)}{6 \left(\iota _3-\iota _4\right)} -\frac{t \left(3 \iota _3^2-5 \iota _4^2\right)^3}{216 \left(\iota _3-\iota _4\right)^3}} \left(\cos \left(y \rho _2\right) \Theta _1+\sin \left(y \rho _2\right) \Theta _2\right)\\&+&e^{x \iota _3+t \varsigma _3} [\cos \left(x \iota _4+t \varsigma _4\right) \Theta _3+\sin \left(x \iota _4+t \varsigma _4\right) \Theta _4]]. \end{eqnarray}$

(2.37)

The dynamical behavior to Eq. (38) is demonstrated in Figure 11.

Figure 11.

$\iota_3 = k_2 = k_1 = \Theta_1 = 1$ ,

$\iota_4 = -2$ ,

$\rho_2 = \Theta_3 = \Theta_4 = 1$ , (a)

$x = -5$ , (b)

$x = 0$ and (c)

$x = 5$ .

DownLoad: Full-Size Img PowerPoint

Case (12)

$\begin{eqnarray} \rho_1 = \rho_4, \iota_2 = \iota_4 = \varsigma_4 = \varsigma_2 = 0, \iota_1 = \iota_3, \varsigma_3 = -\iota _3^3, \varsigma_1 = -\iota _1^3, \end{eqnarray}$

(2.38)

$\begin{eqnarray} \Psi & = & e^{2 \left(-t \iota _3^3+x \iota _3+y \rho _4\right)} k_1+e^{2 y \rho _4} k_2+e^{-t \iota _3^3+x \iota _3+y \rho _4} [\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2]+e^{-t \iota _3^3+x \iota _3+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.39)

$\begin{eqnarray} u_{12} & = & -[2 [2 e^{2 \left(-t \iota _3^3+x \iota _3+y \rho _4\right)} k_1 \iota _3+e^{-t \iota _3^3+x \iota _3+y \rho _4} [\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2] \iota _3+e^{-t \iota _3^3+x \iota _3+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4] \iota _3]]\\&/&[e^{2 \left(-t \iota _3^3+x \iota _3+y \rho _4\right)} k_1+e^{2 y \rho _4} k_2+e^{-t \iota _3^3+x \iota _3+y \rho _4} [\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2]+e^{-t \iota _3^3+x \iota _3+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.40)

The dynamical behavior to Eq. (41) is demonstrated in Figure 12.

Figure 12.

$\iota_3 = k_2 = k_1 = \Theta_1 = 1$ ,

$\iota_4 = -2$ ,

$\rho_2 = \rho_3 = \rho_4 = \Theta_3 = \Theta_4 = 1$ ,

$\Theta_2 = -1$ , (a)

$x = -10$ , (b)

$x = 0$ and (c)

$x = 10$ .

DownLoad: Full-Size Img PowerPoint

Case (13)

$\begin{eqnarray} \rho_1 = \rho_4, \rho_2 = \iota_3 = \iota_1 = \varsigma_4 = \iota_4 = \varsigma_1 = 0, \varsigma_3 = -\iota _3^3, \varsigma_2 = \iota _2^3, \end{eqnarray}$

(2.41)

$\begin{eqnarray} \Psi & = & e^{2 y \rho _4} k_1+e^{2 y \rho _4} k_2+e^{y \rho _4} [\cos \left(t \iota _2^3+x \iota _2\right) \Theta _1+\sin \left(t \iota _2^3+x \iota _2\right) \Theta _2]\\&+&e^{y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.42)

$\begin{eqnarray} u_{13} & = & -[2 e^{y \rho _4} [\cos \left(t \iota _2^3+x \iota _2\right) \iota _2 \Theta _2-\sin \left(t \iota _2^3+x \iota _2\right) \iota _2 \Theta _1]]/[e^{2 y \rho _4} k_1\\&+&e^{2 y \rho _4} k_2+e^{y \rho _4} [\cos \left(t \iota _2^3+x \iota _2\right) \Theta _1+\sin \left(t \iota _2^3+x \iota _2\right) \Theta _2]\\&+&e^{y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.43)

The dynamical behavior to Eq. (44) is demonstrated in Figure 13.

Figure 13.

$\iota_2 = k_2 = k_1 = \Theta_1 = 1$ ,

$\Theta_3 = \Theta_4 = 1$ ,

$\rho_3 = \rho_4 = \Theta_2 = -1$ , (a)

$t = -10$ , (b)

$t = 0$ and (c)

$t = 10$ .

DownLoad: Full-Size Img PowerPoint

Case (14)

$\begin{eqnarray} \rho_1& = &\rho_4, \rho_2 = \Theta_3 = \Theta_4 = \iota_4 = \varsigma_4 = 0,\\ \varsigma_1& = &3 \iota _1 \iota _2^2-\iota _1^3, \varsigma_2 = \iota _2 \left(\iota _2^2-3 \iota _1^2\right), \end{eqnarray}$

(2.44)

$\begin{eqnarray} \Psi & = & e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4]} k_1+e^{2 y \rho _4} k_2+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4} [\cos [x \iota _2\\&+&t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _1+\sin [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _2]. \end{eqnarray}$

(2.45)

$\begin{eqnarray} u_{14} & = & -[2 [2 e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4]} k_1 \iota _1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4} [\cos [x \iota _2\\&+&t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _1+\sin [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \Theta _2] \iota _1\\&+&e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4} [\cos [x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \iota _2 \Theta _2-\sin [x \iota _2\\&+&t \left(\iota _2^2-3 \iota _1^2\right) \iota _2] \iota _2 \Theta _1]]]/[e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4]} k_1+e^{2 y \rho _4} k_2\\&+&e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _4} [\cos \left(x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2\right) \Theta _1\\&+&\sin \left(x \iota _2+t \left(\iota _2^2-3 \iota _1^2\right) \iota _2\right) \Theta _2]]. \end{eqnarray}$

(2.46)

The dynamical behavior to Eq. (47) is demonstrated in Figure 14.

Figure 14.

$\iota_2 = k_2 = k_1 = \iota_1 = 1$ ,

$\Theta_1 = 1$ ,

$\rho_4 = \Theta_2 = -1$ , (a)

$x = -5$ , (b)

$x = 0$ and (c)

$x = 5$ .

DownLoad: Full-Size Img PowerPoint

Case (15)

$\begin{eqnarray} k_2& = &\iota_4 = \varsigma_4 = \rho_2 = 0, \varsigma_2 = \iota _2^3-3 \iota _1^2 \iota _2, \varsigma_1 = 3 \iota _1 \iota _2^2-\iota _1^3, \iota_3 = \iota _1+i \iota _2,\\ \Theta_2& = &i \Theta _1, \Theta_4 = \frac{\left(\rho _1-\rho _3\right) \Theta _3}{\rho _4}, \varsigma_3 = 6 \iota _1^3-12 \iota _3 \iota _1^2+6 \left(\iota _2^2+\iota _3^2\right) \iota _1-\iota _3^3, \end{eqnarray}$

(2.47)

$\begin{eqnarray} \Psi & = & e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1]} k_1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2+t (\iota _2^3\\&-&3 \iota _1^2 \iota _2)] \Theta _1+i \sin \left(x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)\right) \Theta _1]+e^{x \left(\iota _1+i \iota _2\right)+y \rho _3+t \varsigma _3}\\&*& [\cos \left(y \rho _4\right) \Theta _3+\frac{\sin \left(y \rho _4\right) \left(\rho _1-\rho _3\right) \Theta _3}{\rho _4}]. \end{eqnarray}$

(2.48)

$\begin{eqnarray} u_{15} & = & -[2 [2 e^{2 [x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1]} k_1 \iota _1+e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [\cos [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _1+i \sin \left(x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)\right) \Theta _1] \iota _1\\&+&e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} [i \cos [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \iota _2 \Theta _1-\sin [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \iota _2 \Theta _1]+e^{x \left(\iota _1+i \iota _2\right)+y \rho _3+t \varsigma _3} \left(\iota _1+i \iota _2\right) [\cos \left(y \rho _4\right) \Theta _3\\&+&\frac{\sin \left(y \rho _4\right) \left(\rho _1-\rho _3\right) \Theta _3}{\rho _4}]]]/[e^{2 \left(x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1\right)} k_1+[\cos [x \iota _2\\&+&t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _1+i \sin [x \iota _2+t \left(\iota _2^3-3 \iota _1^2 \iota _2\right)] \Theta _1]\\&*& e^{x \iota _1+t \left(3 \iota _1 \iota _2^2-\iota _1^3\right)+y \rho _1} +e^{x \left(\iota _1+i \iota _2\right)+y \rho _3+t \varsigma _3}\\&*& [\cos \left(y \rho _4\right) \Theta _3+\frac{\sin \left(y \rho _4\right) \left(\rho _1-\rho _3\right) \Theta _3}{\rho _4}]]. \end{eqnarray}$

(2.49)

Case (16)

$\begin{eqnarray} k_1& = &\rho_4 = 0, \Theta_1 = i \Theta _2, \varsigma_4 = \iota _4 \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right), \rho_1 = i \rho _2+\rho _3,\\ \varsigma_3& = &[-2 \Theta _3 \iota _3^3-6 \iota _4 \left(\Theta _4-2 \Theta _3\right) \iota _3^2-6 \iota _4^2 \left(3 \Theta _3-4 \Theta _4\right) \iota _3+2 \iota _4^3 \left(2 \Theta _3-11 \Theta _4\right)\\&+&\left(2 \Theta _3-\Theta _4\right) \left(\left(\iota _1-i \iota _2-2 \iota _4\right)^3+\varsigma _1-i \varsigma _2\right)]/(2 \Theta _3),\\ \varsigma_1& = &-\iota _1^3+6 \iota _4 \iota _1^2+3 \iota _2^2 \iota _1-12 \iota _4^2 \iota _1-14 \iota _4^3+24 \iota _3 \iota _4^2-6 \iota _2^2 \iota _4-6 \iota _3^2 \iota _4\\&+&i \left(-\iota _2^3+3 \iota _1^2 \iota _2+12 \iota _4^2 \iota _2-12 \iota _1 \iota _4 \iota _2+\varsigma _2\right), \end{eqnarray}$

(2.50)

$\begin{eqnarray} \Psi & = & e^{2 \left(x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4\right)} k_2+e^{x \iota _1+y \left(i \rho _2+\rho _3\right)+t \varsigma _1} [i \cos (x \iota _2+y \rho _2\\&+&t \varsigma _2) \Theta _2+\sin \left(x \iota _2+y \rho _2+t \varsigma _2\right) \Theta _2]+e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos [x \iota _4+t (-3 \iota _3^2+12\\&*& \iota _4 \iota _3-11 \iota _4^2) \iota _4] \Theta _3+\sin \left(x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4\right) \Theta _4]. \end{eqnarray}$

(2.51)

$\begin{eqnarray} u_{16} & = & -[2 [2 e^{2 [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2 \iota _4+e^{x \iota _1+y \left(i \rho _2+\rho _3\right)+t \varsigma _1} \iota _1 [i \cos \\&&\left(x \iota _2+y \rho _2+t \varsigma _2\right) \Theta _2 +\sin \left(x \iota _2+y \rho _2+t \varsigma _2\right) \Theta _2]+e^{x \iota _1+y \left(i \rho _2+\rho _3\right)+t \varsigma _1}\\&*& [\cos \left(x \iota _2 +y \rho _2+t \varsigma _2\right) \iota _2 \Theta _2-i \sin \left(x \iota _2+y \rho _2+t \varsigma _2\right) \iota _2 \Theta _2]\\&+&e^{x \iota _3+y \rho _3 +t \varsigma _3} \iota _3 [\cos [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _3+ \sin [x \iota _4\\&+&t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]) \Theta _4]+e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos [x \iota _4+t (-3 \iota _3^2\\&+&12 \iota _4 \iota _3-11 \iota _4^2) \iota _4] \iota _4 \Theta _4-\sin [x \iota _4 +t (-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2) \iota _4] \iota _4\\&*& \Theta _3]]]/[e^{2 [x \iota _4+t \left(-3 \iota _3^2 +12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4]} k_2+e^{x \iota _1+y \left(i \rho _2+\rho _3\right)+t \varsigma _1} [i \cos (x \iota _2 \\&+&y \rho _2+t \varsigma _2) \Theta _2+\sin \left(x \iota _2+y \rho _2+t \varsigma _2\right) \Theta _2]+e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos [x \iota _4\\&+&t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _3\\&+&\sin [x \iota _4+t \left(-3 \iota _3^2+12 \iota _4 \iota _3-11 \iota _4^2\right) \iota _4] \Theta _4]]. \end{eqnarray}$

(2.52)

Case (17)

$\begin{eqnarray} \rho_1& = &\rho_4, \iota_2 = \varsigma_2 = 0, \Theta_4 = i \Theta _3, \varsigma_1 = -\left(\iota _3+i \iota _4\right)^3, \iota_1 = \iota _3+i \iota _4,\\ \varsigma_4& = &-4 \iota _4^3+6 \iota _1 \iota _4^2-3 \iota _1^2 \iota _4, \varsigma_3 = -\iota _3^3-(3+6 i) \iota _4^2 \iota _3+(6+2 i) \iota _4^3, \end{eqnarray}$

(2.53)

$\begin{eqnarray} \Psi & = & e^{2 \left(-t \left(\iota _3+i \iota _4\right)^3+x \left(\iota _3+i \iota _4\right)+y \rho _4\right)} k_1+e^{2 \left(x \iota _4+y \rho _4+t \varsigma _4\right)} k_2+[\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2] e^{-t \left(\iota _3+i \iota _4\right)^3+x \left(\iota _3+i \iota _4\right)+y \rho _4} +[\cos \left(x \iota _4+y \rho _4+t \varsigma _4\right) \Theta _3\\&+&i \sin \left(x \iota _4+y \rho _4+t \varsigma _4\right) \Theta _3] e^{x \iota _3+t \left(-\iota _3^3-(3+6 i) \iota _4^2 \iota _3+(6+2 i) \iota _4^3\right)+y \rho _3}. \end{eqnarray}$

(2.54)

$\begin{eqnarray} u_{17} & = & -[2 [2 e^{2 [-t \left(\iota _3+i \iota _4\right)^3+x \left(\iota _3+i \iota _4\right)+y \rho _4]} k_1 \left(\iota _3+i \iota _4\right)+[\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2] \left(\iota _3+i \iota _4\right) e^{-t \left(\iota _3+i \iota _4\right)^3+x \left(\iota _3+i \iota _4\right)+y \rho _4}+2 e^{2 \left(x \iota _4+y \rho _4+t \varsigma _4\right)} k_2\\&*& \iota _4+e^{x \iota _3+t \left(-\iota _3^3-(3+6 i) \iota _4^2 \iota _3+(6+2 i) \iota _4^3\right)+y \rho _3} \iota _3 [\cos \left(x \iota _4+y \rho _4+t \varsigma _4\right) \Theta _3\\&+&i \sin \left(x \iota _4+y \rho _4+t \varsigma _4\right) \Theta _3]+e^{x \iota _3+t \left(-\iota _3^3-(3+6 i) \iota _4^2 \iota _3+(6+2 i) \iota _4^3\right)+y \rho _3}\\&*& [i \cos \left(x \iota _4+y \rho _4+t \varsigma _4\right) \iota _4 \Theta _3-\sin \left(x \iota _4+y \rho _4+t \varsigma _4\right) \iota _4 \Theta _3]]]\\&/&[e^{2 [-t \left(\iota _3+i \iota _4\right)^3+x \left(\iota _3+i \iota _4\right)+y \rho _4]} k_1+e^{2 \left(x \iota _4+y \rho _4+t \varsigma _4\right)} k_2+[\cos \left(y \rho _2\right) \Theta _1\\&+&\sin \left(y \rho _2\right) \Theta _2] e^{-t \left(\iota _3+i \iota _4\right)^3+x \left(\iota _3+i \iota _4\right)+y \rho _4}+[\cos \left(x \iota _4 +y \rho _4+t \varsigma _4\right) \Theta _3\\&+&i \sin \left(x \iota _4+y \rho _4+t \varsigma _4\right) \Theta _3] e^{x \iota _3+t \left(-\iota _3^3-(3+6 i) \iota _4^2 \iota _3+(6+2 i) \iota _4^3\right)+y \rho _3}]. \end{eqnarray}$

(2.55)

Case (18)

$\begin{eqnarray} \rho_1& = &\rho_4, \rho_2 = \iota_1 = \iota_4 = \varsigma_4 = \varsigma_1 = 0, \varsigma_2 = \iota_2^3,\\ \varsigma_3& = &-\iota_3^3, \Theta_1 = -\frac{\iota _3 \Theta _2}{\iota _2}, \iota_3 = i \iota_2, \end{eqnarray}$

(2.56)

$\begin{eqnarray} \Psi & = & e^{2 y \rho _4} k_1+e^{2 y \rho _4} k_2+e^{y \rho _4} [\sin \left(t \iota _2^3+x \iota _2\right) \Theta _2-i \cos \left(t \iota _2^3+x \iota _2\right) \Theta _2]\\&+&e^{i t \iota _2^3+i x \iota _2+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]. \end{eqnarray}$

(2.57)

$\begin{eqnarray} u_{18} & = & -[2 [e^{y \rho _4} [\cos \left(t \iota _2^3+x \iota _2\right) \iota _2 \Theta _2+i \sin \left(t \iota _2^3+x \iota _2\right) \iota _2 \Theta _2]\\&+&i e^{i t \iota _2^3+i x \iota _2+y \rho _3} \iota _2 [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]]/[e^{2 y \rho _4} k_1+e^{2 y \rho _4} k_2\\&+&e^{y \rho _4} \left(\sin \left(t \iota _2^3+x \iota _2\right) \Theta _2-i \cos \left(t \iota _2^3+x \iota _2\right) \Theta _2\right)\\&+&e^{i t \iota _2^3+i x \iota _2+y \rho _3} [\cos \left(y \rho _4\right) \Theta _3+\sin \left(y \rho _4\right) \Theta _4]]. \end{eqnarray}$

(2.58)

Case (19)

$\begin{eqnarray} \rho_1& = &\rho_4 = \rho_2 = 0, \iota_2 = i \iota_1, \iota_4 = \iota_3-\iota_1, \Theta_3 = -\frac{3 \iota _1 \Theta _4}{5 \iota _1-2 \iota _3}, \Theta_1 = i \Theta_2,\\ \varsigma_1& = & \left(3 \iota _1-2 \iota _3\right) (3 \iota _1^2-6 \iota _3 \iota _1+2 \iota _3^2), \varsigma_3 = 20 \iota _1^3-45 \iota _3 \iota _1^2+30 \iota _3^2 \iota _1\\&-&6 \iota _3^3, \varsigma_4 = \left(\iota _1-\iota _3\right) \left(11 \iota _1^2-10 \iota _3 \iota _1+2 \iota _3^2\right),\\ \varsigma_2& = &i \left(3 \iota _1-2 \iota _3\right) \left(3 \iota _1^2-6 \iota _3 \iota _1+2 \iota _3^2\right), \end{eqnarray}$

(2.59)

$\begin{eqnarray} \Psi & = & e^{2 \left(x \iota _1+t \varsigma _1\right)} k_1+e^{2 [x \left(\iota _3-\iota _1\right)+t \varsigma _4]} k_2+e^{x \iota _1+t \varsigma _1} [\cosh \left(x \iota _1-i t \varsigma _2\right) \Theta _1\\&+&i \sinh \left(x \iota _1-i t \varsigma _2\right) \Theta _2]+ e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos \left(x \left(\iota _3-\iota _1\right)+t \varsigma _4\right) \Theta _3\\&+&\sin \left(x \left(\iota _3-\iota _1\right) +t \varsigma _4\right) \Theta _4]. \end{eqnarray}$

(2.60)

$\begin{eqnarray} u_{19} & = & -[2 [2 e^{2 \left(x \iota _1+t \varsigma _1\right)} k_1 \iota _1+e^{x \iota _1+t \varsigma _1} [\cosh \left(x \iota _1-i t \varsigma _2\right) \Theta _1+i \sinh (x \iota _1\\&-&i t \varsigma _2) \Theta _2] \iota _1+2 e^{2 \left(x \left(\iota _3-\iota _1\right)+t \varsigma _4\right)} k_2 \left(\iota _3-\iota _1\right)+e^{x \iota _1+t \varsigma _1} [\sinh \left(x \iota _1-i t \varsigma _2\right)\\&*& \iota _1 \Theta _1+i \cosh \left(x \iota _1-i t \varsigma _2\right) \iota _1 \Theta _2]+e^{x \iota _3+y \rho _3+t \varsigma _3} \iota _3 [\cos [x \left(\iota _3-\iota _1\right)\\&+&t \varsigma _4] \Theta _3+\sin [x \left(\iota _3-\iota _1\right)+t \varsigma _4] \Theta _4]+e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos [x \left(\iota _3-\iota _1\right)\\&+&t \varsigma _4] \left(\iota _3-\iota _1\right) \Theta _4-\sin [x \left(\iota _3-\iota _1\right)+t \varsigma _4] \left(\iota _3-\iota _1\right) \Theta _3]]]/[e^{2 \left(x \iota _1+t \varsigma _1\right)} k_1\\&+&e^{2 \left(x \left(\iota _3-\iota _1\right)+t \varsigma _4\right)} k_2+e^{x \iota _1+t \varsigma _1} [\cosh \left(x \iota _1-i t \varsigma _2\right) \Theta _1+i \sinh \left(x \iota _1-i t \varsigma _2\right)\\&*& \Theta _2]+e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos \left(x \left(\iota _3-\iota _1\right)+t \varsigma _4\right) \Theta _3\\&+&\sin \left(x \left(\iota _3-\iota _1\right)+t \varsigma _4\right) \Theta _4]]. \end{eqnarray}$

(2.61)

Case (20)

$\begin{eqnarray} \rho_1& = &\rho_4 = \rho_2 = 0, \iota_2 = i \iota_1, \iota_4 = \iota_1, \Theta_3 = -\frac{3 \left(\iota _1-\iota _3\right) \Theta _4}{5 \iota _1-3 \iota _3}, \Theta_1 = i \Theta_2,\\ \varsigma_1& = & \frac{-11 \Theta _4 \iota _1^3+12 \iota _3 \Theta _4 \iota _1^2-3 \iota _3^2 \Theta _4 \iota _1}{\Theta _4}, \varsigma_2 = -i \iota _1 \left(11 \iota _1^2-12 \iota _3 \iota _1+3 \iota _3^2\right), \\ \varsigma_4& = &\iota _1 \left(-11 \iota _1^2+12 \iota _3 \iota _1-3 \iota _3^2\right), \varsigma_3 = -20 \iota _1^3+15 \iota _3 \iota _1^2-\iota _3^3, \end{eqnarray}$

(2.62)

$\begin{eqnarray} \Psi & = & e^{2 \left(x \iota _1+t \varsigma _1\right)} k_1+e^{2 \left(x \iota _1+t \varsigma _4\right)} k_2+e^{x \iota _1+t \varsigma _1} [\cosh \left(x \iota _1-i t \varsigma _2\right) \Theta _1\\&+&i \sinh \left(x \iota _1-i t \varsigma _2\right) \Theta _2]+e^{x \iota _3 +y \rho _3+t \varsigma _3} [\cos \left(x \iota _1+t \varsigma _4\right) \Theta _3\\&+&\sin \left(x \iota _1+t \varsigma _4\right) \Theta _4]. \end{eqnarray}$

(2.63)

$\begin{eqnarray} u_{20} & = & -[2 [2 e^{2 \left(x \iota _1+t \varsigma _1\right)} k_1 \iota _1+2 e^{2 \left(x \iota _1+t \varsigma _4\right)} k_2 \iota _1+e^{x \iota _1+t \varsigma _1} [\cosh \left(x \iota _1-i t \varsigma _2\right) \Theta _1\\&+&i \sinh \left(x \iota _1-i t \varsigma _2\right) \Theta _2] \iota _1+e^{x \iota _1+t \varsigma _1} [\sinh \left(x \iota _1-i t \varsigma _2\right) \iota _1 \Theta _1+i \cosh (x \iota _1\\&-&i t \varsigma _2) \iota _1 \Theta _2]+e^{x \iota _3+y \rho _3+t \varsigma _3} \iota _3 [\cos \left(x \iota _1+t \varsigma _4\right) \Theta _3+\sin \left(x \iota _1 +t \varsigma _4\right) \Theta _4]\\&+&e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos \left(x \iota _1+t \varsigma _4\right) \iota _1 \Theta _4-\sin \left(x \iota _1 +t \varsigma _4\right) \iota _1 \Theta _3]]]/[e^{2 \left(x \iota _1+t \varsigma _1\right)}\\&*& k_1+e^{2 \left(x \iota _1+t \varsigma _4\right)} k_2+ e^{x \iota _1+t \varsigma _1} [\cosh \left(x \iota _1-i t \varsigma _2\right) \Theta _1+i \sinh \left(x \iota _1-i t \varsigma _2\right)\\&*& \Theta _2] +e^{x \iota _3+y \rho _3+t \varsigma _3} [\cos \left(x \iota _1+t \varsigma _4\right) \Theta _3+\sin \left(x \iota _1+t \varsigma _4\right) \Theta _4]]. \end{eqnarray}$

(2.64)

3. Conclusion

In this paper, the (2+1)-dimensional BLMP equation is discussed, which describes the incompressible fluid. Based on the bilinear form and an ansätz functions, many entirely new complexiton solutions and double periodic-soliton solutions are obtained. The dynamical behaviors are demonstrated in some three-dimensional plots by setting different values of the parameters. The ansätz function is very effective in solving the periodic solutions and complexiton solutions of the higher order nonlinear evolution equations. In Figs. 1-14, it is obvious that the waves are repeated at intervals of time or distance. All the solutions have been verified to be correct by symbolic computation software Mathematica.

Acknowledgments

We would like to express our sincere thanks to the reviewers and editors for their valuable suggestions. Project supported by National Natural Science Foundation of China (Grant No 81960715).

Conflict of interest

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

Symbolic program

$\begin{eqnarray} &&In[1]: = \Psi[x,y,t] = Exp[\iota_1\,x+\rho_1\,y+\varsigma_1\,t] [\Theta_1 \cos[\iota_2\,x+\rho_2\,y+\varsigma_2\,t]\\&&+\Theta_2 \sin[\iota_2\,x+\rho_2\,y+\varsigma_2\,t]]+k_1 Exp[2 (\iota_1\,x+\rho_1\,y+\varsigma_1\,t)]\\&&+Exp[\iota_3\,x+\rho_3\,y+\varsigma_3\,t] [\Theta_3 \cos[\iota_4\,x+\rho_4\,y+\varsigma_4\,t]\\&&+\Theta_4 \sin[\iota_4\,x+\rho_4\,y+\varsigma_4\,t]]+k_2 Exp[\iota_4\,x+\rho_4\,y+\varsigma_4\,t]\\ &&In[2]: = M = Expand[-\Psi_t \Psi_y-\Psi_{xxx} \Psi_y+3 \Psi_{xy} \Psi_{xx}\\&&-3 \Psi_x \Psi_{xxy}+\Psi \left(\Psi_{yt}+\Psi_{xxxy}\right)]\\ &&In[3]: = M1 = FullSimplify[Coefficient[M,Exp[2 (\iota_1\,x+\rho_1\,y+\varsigma_1\,t)\\&&+2 (\iota_4\,x+\rho_4\,y+\varsigma_4\,t)]]]\\ &&In[4]: = M2 = FullSimplify[Coefficient[M,Exp[3 (\iota_1\,x+\rho_1\,y+\varsigma_1\,t)]]]\\ &&In[5]: = M3 = FullSimplify[Coefficient[M,Sin[\iota_2\,x+\rho_2\,y+\varsigma_2\,t]]]\\ &&In[6]: = M4 = FullSimplify[Coefficient[M,Cos[\iota_2\,x+\rho_2\,y+\varsigma_2\,t]]]\\ &&In[7]: = M5 = FullSimplify[Coefficient[M,Sin[\iota_4\,x+\rho_4\,y+\varsigma_4\,t]]]\\ &&In[8]: = M6 = FullSimplify[Coefficient[M,Cos[\iota_4\,x+\rho_4\,y+\varsigma_4\,t]]] \end{eqnarray}$

References

[1]	T. Motoda. Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions, AIMS Mathematics, 3 (2018), 263-287.
[2]	A. M. Wazwaz. A variety of negative-order integrable KdV equations of higher orders, Wave. Random. Complex., 29 (2019), 195-203.
[3]	W. P. Gao, Y. X. Hu. The exact traveling wave solutions of a class of generalized Black-Scholes equation, AIMS Mathematics, 2 (2017), 385-399.
[4]	Z. F. Zeng, J. G. Liu, Y. Jiang, et al. Transformations and soliton solutions for a variable-coefficient nonlinear schrödinger equation in the dispersion decreasing fiber with symbolic computation, Fund. Inform., 145 (2016), 207-219. doi: 10.3233/FI-2016-1355
[5]	M. T. Islam, M. A. Akbar, M. A. Azad. Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative, AIMS Mathematics, 3 (2018), 625-646.
[6]	W. X. Ma, X. Yong, H. Q. Zhang. Diversity of interaction solutions to the (2+1)-dimensional ito equation, Comput. Math. Appl., 75 (2018), 289-295.
[7]	A. Bashan. An Efficient Approximation to Numerical Solutions for the Kawahara Equation Via Modified Cubic B-Spline Differential Quadrature Method, Mediterr. J. Math., 16 (2019), 14.
[8]	A. Bashan. A novel approach via mixed Crank-Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation, Pramana, 92 (2019), 84.
[9]	A. Bashan. A mixed algorithm for numerical computation of soliton solutions of the coupled KdV equation: Finite difference method and differential quadrature method, Appl. Math. Comput., 360 (2019), 42-57.
[10]	M. S. Osman. On complex wave solutions governed by the 2d Ginzburg-Landau equation with variable coefficients, Optik, 156 (2018), 169-174.
[11]	Y. H. Yin, W. X. Ma, J. G. Liu, et al. Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction, Comput. Math. Appl., 76 (2018), 1275-1283. doi: 10.1016/j.camwa.2018.06.020
[12]	Y. Kong, L. Xin, Q. Qiu, et al. Exact periodic wave solutions for the modified Zakharov equations with a quantum correction, Appl. Math. Lett., 94 (2019), 140-148. doi: 10.1016/j.aml.2019.01.009
[13]	Y. Z. Li, J. G. Liu. Multiple periodic-soliton solutions of the (3 + 1)-dimensional generalised shallow water equation, Pramana, 90 (2018), 71.
[14]	W. X. Ma, Y. Zhou. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differ. Equations., 264 (2018), 2633-2659.
[15]	G. Akram, F. Batool. Solitary wave solutions of the Schafer-Wayne short-pulse equation using two reliable methods, Opt. Quant. Electron., 49 (2017), 14.
[16]	J. Y. Yang, W. X. Ma, Z. Y. Qin. Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation, Anal. Math. Phys., 8 (2018), 427-436.
[17]	Z. Z. Lan. Rogue wave solutions for a coupled nonlinear Schrödinger equation in the birefringent optical fiber, Appl. Math. Lett., 98 (2019), 128-134.
[18]	L. N. Gao, X. Y. Zhao, Y. Y. Zi, et al. esonant behavior of multiple wave solutions to a Hirota bilinear equation, Comput. Math. Appl., 72 (2016), 1225-1229. doi: 10.1016/j.camwa.2016.06.008
[19]	A. M. Wazwaz. A study on a (2+1)-dimensional and a (3+1)-dimensional generalized Burgers equation, Appl. Math. Lett., 31 (2014), 41-45.
[20]	J. G. Liu. Double-periodic soliton solutions for the (3+1)-dimensional Boiti-Leon-MannaPempinelli equation in incompressible fluid, Comput. Math. Appl., 75 (2018), 3604-3613.
[21]	Y. F. Hua, B. L. Guo, W. X. Ma, et al. Interaction behavior associated with a generalized (2+1)- dimensional Hirota bilinear equation for nonlinear waves, Appl. Math. Model., 74 (2019), 184-198. doi: 10.1016/j.apm.2019.04.044
[22]	X. P. Zeng, Z. D. Dai, D. L. Li. New periodic soliton solutions for the (3 + 1)-dimensional potentialYTSF equation, Chaos. Soliton. Fract., 42 (2009), 657-661.
[23]	O. Gonzalez-Gaxiola. Bright and dark optical solitons of the Schafer-Wayne short-pulse equation by Laplace substitution method, Optik, 200 (2020), 163414.
[24]	J. G. Liu, J. Q. Du, Z. F. Zeng, et al. Exact periodic cross-kink wave solutions for the new (2+1) dimensional kdv equation in fluid flows and plasma physics, Chaos, 26 (2016), 989-1002.
[25]	A. M. Wazwaz. Multiple complex and multiple real soliton solutions for the integrable sine-Gordon equation, Optik, 172 (2018), 622-627.
[26]	Y. N. Tang, W. J. Zai. New periodic-wave solutions for (2+1)- and (3+1)-dimensional Boiti-LeonManna-Pempinelli equations, Nonlinear Dyn., 81 (2015), 249-255.
[27]	L. Luo. New exact solutions and Bäklund transformation for Boiti-Leon-Manna-Pempinelli equation, Phys. Lett. A., 375 (2001), 1059-1063.
[28]	S. H. Ma, J. P. Fang. Multi dromion-solitoff and fractal excitations for (2+1)-dimensional BoitiLeon-Manna-Pempinelli system, Commun. Theor. Phys., 52 (2009), 641-645.
[29]	L. Delisle, M. Mosaddeghi. Classical and SUSY solutions of the Boiti-Leon-Manna-Pempinelli equation, J. Phys. A Math. Theor., 46 (2013), 115203.
[30]	M. Najafi, M. Najafi, S. Arbabi. Wronskian determinant solutions of the (2+1)-dimensional BoitiLeon-Manna-Pempinelli equation, Int. J. Adv. Math. Sci., 1 (2013), 8-11.
[31]	Z. H. Fu, J. G. Liu. Exact periodic cross-kink wave solutions for the (2+1)-dimensional Boiti-LeonManna-Pempinelli equation, Indian. J. Pure. Ap. Phy., 55 (2017), 163-167.
[32]	Y. Li, D. S. Li. New exact solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Appl. Math. Sci., 6 (2012), 579-587.
[33]	K. Melike, A. Arzu, B. Ahmet. The Auto-Bäcklund transformations for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, AIP Conference Proceedings, 1798 (2017), 020071.
[34]	K. Melike. Two different systematic techniques to find analytical solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Chinese J. Phys., 56 (2018), 2523-2530.
[35]	C. J. Bai, H. Zhao. New solitary wave and jacobi periodic wave excitations in (2+1)-demensional Boiti-Leon-Manna-Pempinelli system, Int. J. Mod. Phys. B, 22 (2008), 2407-2420.

This article has been cited by:

1.	Chen Yue, Mostafa M. A. Khater, Mustafa Inc, Raghda A. M. Attia, Dianchen Lu, Abundant analytical solutions of the fractional nonlinear (2 + 1)-dimensional BLMP equation arising in incompressible fluid, 2020, 34, 0217-9792, 2050084, 10.1142/S0217979220500848
2.	Kalim U. Tariq, Aly R. Seadawy, Syed T. R. Rizvi, Rizwan Javed, Some optical soliton solutions to the generalized (1 + 1)-dimensional perturbed nonlinear Schrödinger equation using two analytical approaches, 2022, 36, 0217-9792, 10.1142/S0217979222501776
3.	Chun-Rong Qin, Jian-Guo Liu, Study on double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients, 2022, 34, 22113797, 105254, 10.1016/j.rinp.2022.105254
4.	Ali Althobaiti, Saad Althobaiti, K. El-Rashidy, Aly R. Seadawy, Exact solutions for the nonlinear extended KdV equation in a stratified shear flow using modified exponential rational method, 2021, 29, 22113797, 104723, 10.1016/j.rinp.2021.104723
5.	Litao Gai, Wen-Xiu Ma, Bilige Sudao, Abundant multilayer network model solutions and bright-dark solitons for a (3 + 1)-dimensional p-gBLMP equation, 2021, 106, 0924-090X, 867, 10.1007/s11071-021-06864-8
6.	Jiajia Yang, Meng Jin, Xiangpeng Xin, Lie symmetry analysis, optimal system and exact solutions for variable-coefficients Boiti-Leon-Manna-Pempinelli equation, 2024, 99, 0031-8949, 025233, 10.1088/1402-4896/ad1a32
7.	Wen-Hui Zhu, Jian-Guo Liu, Mohammad Asif Arefin, M. Hafiz Uddin, Ya-Kui Wu, Mohammad Mirzazadeh, Double-Periodic Soliton Solutions of the (2+1)-Dimensional Ito Equation, 2023, 2023, 1687-9139, 1, 10.1155/2023/9321673
8.	Handenur Esen, Aydin Secer, Muslum Ozisik, Mustafa Bayram, Obtaining soliton solutions of the nonlinear (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation via two analytical techniques, 2024, 38, 0217-9792, 10.1142/S0217979224500103
9.	Wenfang Li, Yingchun Kuang, Jalil Manafian, Somaye Malmir, Baharak Eslami, K. H. Mahmoud, A. S. A. Alsubaie, Multiple rogue wave, double-periodic soliton and breather wave solutions for a generalized breaking soliton system in (3 + 1)-dimensions, 2024, 14, 2045-2322, 10.1038/s41598-024-70523-2

Reader Comments

Your name:*

Email:*
© 2020 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(4475) PDF downloads(389) Cited by(9)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(14)

AIMS Mathematics

Complexiton solutions and periodic-soliton solutions for the (2+1)-dimensional BLMP equation

Related Papers:

Abstract

1. Introduction

2. Complexiton solutions and double periodic-soliton solutions

3. Conclusion

Acknowledgments

Conflict of interest

Symbolic program

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

Complexiton solutions and periodic-soliton solutions for the (2+1)-dimensional BLMP equation

Related Papers:

Abstract

1. Introduction

2. Complexiton solutions and double periodic-soliton solutions

3. Conclusion

Acknowledgments

Conflict of interest

Symbolic program

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