A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations

Chun-Ku Kuo; Dipankar Kumar; Chieh-Ju Juan; Chun-Ku Kuo; Dipankar Kumar; Chieh-Ju Juan

doi:10.3934/math.20221136

AIMS Mathematics

2022, Volume 7, Issue 12: 20740-20751. doi: 10.3934/math.20221136

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Research article

A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations

1.
Department of Aeronautics and Astronautics, Air Force Academy, Kaohsiung 820009, Taiwan
2.
Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj 8100, Bangladesh
3.
Department of Aviation Management, Air Force Academy, Kaohsiung 820009, Taiwan

Received: 04 August 2022 Revised: 12 September 2022 Accepted: 19 September 2022 Published: 26 September 2022
MSC : 35C08, 35Q51, 37K40

In this study, a fourth-order nonlinear wave equation with variable coefficients was investigated. Through appropriate choice of the free parameters and using the simplified linear superposition principle (LSP) and velocity resonance (VR), the examined equation can be considered as Hirota–Satsuma–Ito, Calogero–Bogoyavlenskii–Schiff and Jimbo–Miwa equations. The main objective of this study was to obtain novel resonant multi-soliton solutions and investigate inelastic interactions of traveling waves for the above-mentioned equation. Novel resonant multi-soliton solutions along with their essential conditions were obtained by using simplified LSP, and the conditions guaranteed the existence of resonant solitons. Furthermore, the obtained solutions were used to investigate the dynamic and fission behavior of Y-type multi-soliton waves. For an accurate investigation of physical phenomena, appropriate free parameters were chosen to ascertain the impact on the speed of traveling waves and the initiation time of fission. Three-dimensional and contour plots of the obtained solutions are presented in Figures 1–6. Additionally, two nonlinear equations were formulated and investigated using VR, and the related soliton molecules were simultaneously extracted. The reported resonant Y-type multi-soliton waves and equations are new and have not been previously investigated. They can be used to explain modeled physical phenomena and can provide information about dynamic behavior of shallow water waves.

Keywords:

simplified linear superposition principle,
velocity resonance,
soliton molecule,
resonant Y-type multi-soliton

Citation: Chun-Ku Kuo, Dipankar Kumar, Chieh-Ju Juan. A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations[J]. AIMS Mathematics, 2022, 7(12): 20740-20751. doi: 10.3934/math.20221136

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Abstract

1. Introduction

1.1. Overview

Many physical phenomena, in the fields of sound and electrical waves, heat transfer, mechanics, and optics can be described through partial differential equations. Therefore, we can say that the study of mathematical physics depends primarily on the study of partial differential equations. During previous decades, many methods were developed to reduce and solve partial differential equations, including Symmetry method ^[1,2,3,4], scattering transformation ^[5], Tanh method ^[6], the homogeneous balance method ^[7], Aboodh transform decomposition method ^[8], ( $G^{\prime}/G$ , $1/G$ )-expansion method ^[9], Exp-function method ^[10], Generalized He's Exp-function method ^[11], Kudryashov method ^[12], and Generalized Kudryashov method ^[13]. Consider the nonlinear (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation ^[14,15] in the following form:

$\begin{equation} \Phi_{xxxxy}+4\Phi_{xxy}\Phi_{x}+2\Phi_{xxx}\Phi_{y}+6\Phi_{xy}\Phi_{xx} +\Phi_{yyy}-2\Phi_{xxt} = 0, \end{equation}$

(1)

where $\Phi = \Phi(x, y, t).$ DJKM describes the wave diffusion of physical-quantity, and it has certain applications in plasma physics ^[16,17]. This equation was presented by Jimbo and Miwa as second formulas for Kadomtsev-Petviashvili hierarchy.

1.2. Related literatures

There is a wealth of literature related to this governing equation such as: Bibi and Ahmad ^[14] who employed the symmetry method to reduce the governing equation to another equation of lower order and obtained its exact solutions. The exp( $-\phi$ ( $\xi$ ))-Expansion Method was utilized by Sajid and Akram ^[14] for obtaining traveling wave solutions for Eq (1). Halder et al. ^[18] used the Lie symmetry analysis and singularity method to study one type of the governing equation. Wazwaz ^[19] studied the integrability of this equation and obtained multiple soliton solutions by using Painlevé analysis and Hirota's method. Akram et al. ^[16] applied the extended direct algebraic method to obtain exact solutions for the governing equation.

1.3. Main idea

The previous literature led us to determine the study of the governing equation to apply the Lie symmetry method and the Generalized Kudryashov method. Our main objective of this paper is to convert the governing equation to several distinct ordinary differential equations (ODE). We then solve the resulting ODEs to obtain exact and novel wave solutions for the governing equation. As a result, we can show the dynamic behavior of wave solutions in 2-D and 3-D plots.

2. Lie symmetry method

In this part, the Lie symmetry method ^[20,21,22] is used to deduce the similarity reductions for the governing equation, so the one-parameter Lie symmetry of infinitesimal variables is shown as follows:

$\begin{align*} t^{\ast} & = t+\varepsilon A(t,x,y,\Phi)+o(\varepsilon^{2}),\text{ }x^{\ast } = x+\varepsilon B(t,x,y,\Phi)+o(\varepsilon^{2}),\nonumber\\ y^{\ast} & = y+\varepsilon C(t,x,y,\Phi)+o(\varepsilon^{2}),\text{ } \Phi^{\ast} = \Phi+\varepsilon E(t,x,y,\Phi)+o(\varepsilon^{2}), \end{align*}$

(2)

where $\varepsilon$ is the one parameter Lie group, $A, \; B, \; C$ and $E$ are infinitesimal variables and $t, \; x$ and $y$ are independent variables of the function $\Phi.$

The vector generator $\chi$ is related to the above-mentioned group transformations which can be written as the following expression:

$\begin{equation} \chi = A\frac{\partial}{\partial t}+B\frac{\partial}{\partial x}+C\frac {\partial}{\partial y}+E\frac{\partial}{\partial\Phi}, \end{equation}$

(3)

when the following invariance condition is satisfied:

$\begin{equation} \Gamma^{(3)}(\Delta) = 0, \end{equation}$

(4)

where $\Delta$ represents Eq (1) and $\Gamma^{(3)}$ is the third order prolongation^[23,24] of the operator V

$\begin{align*} \Gamma^{(3)} & = \chi+E_{[x]}\frac{\partial}{\partial\Phi_{x}}+E_{[y]} \frac{\partial}{\partial\Phi_{y}}+E_{[xy]}\frac{\partial}{\partial\Phi_{xy} }+E_{[xxx]}\frac{\partial}{\partial\Phi_{xxx}}+E_{[xxy]}\frac{\partial }{\partial\Phi_{xxy}}\nonumber\\ & +E_{[xxt]}\frac{\partial\Phi}{\partial\Phi_{xxt}}+E_{[xxy]}\frac{\partial }{\partial\Phi_{xxy}}+E_{[xxxy]}\frac{\partial}{\partial\Phi_{xxxy}} +E_{[yyy]}\frac{\partial}{\partial\Phi_{yyy}}, \end{align*}$

(5)

where the components $E_{[x]}, E_{[xx]}, \; E_{[xyy]}, \; E_{[y]}, \; E_{[xy]}....$ can be determined from the following expressions:

$\begin{align*} E_{[x]} & = D_{x}E-\Phi_{t}D_{x}A-\Phi_{x}D_{x}B-\Phi_{y}D_{x}C,\nonumber\\ E_{[y]} & = D_{y}E-\Phi_{t}D_{y}A-\Phi_{x}D_{y}B-\Phi_{y}D_{y}C,\nonumber\\ \text{ }E_{[xx]} & = D_{x}E_{[x]}-\Phi_{xt}D_{x}A-\Phi_{xx}D_{x}B-\Phi _{xy}D_{x}C,\nonumber\\ \text{ }E_{[xy]} & = D_{y}E_{[x]}-\Phi_{xt}D_{y}A-\Phi_{xx}D_{y}B-\Phi _{xy}D_{y}C,\nonumber\\ \text{ }E_{[yy]} & = D_{y}E_{[y]}-\Phi_{xt}D_{y}A-\Phi_{xx}D_{y}B-\Phi _{xy}D_{y}C,\nonumber\\ E_{[xxxy]} & = D_{y}E_{[xxx]}-\Phi_{xxxt}D_{y}A-\Phi_{xxxx}D_{y}B-\Phi _{xxxy}D_{y}C. \end{align*}$

(6)

Substituting (1) into Eq (4), yields an identity components $A_{x}, A_{xx}, \; B_{t}, B_{x}, ....$ and adding the coefficients of $\Phi_{x, x, t}$ , $\Phi_{y, y, y}$ , ... and equating them to zero, then the infinitesimals $A, B, C$ and $E$ are obtained from the following equations:

$A_{x} = A_{y} = A_{\Phi} = 0,$

$B_{y} = B_{\Phi} = 0,B_{x} = \frac{1}{4}A_{t},$

$C_{x} = B_{\Phi} = 0,\text{ }C_{z} = \frac{1}{2}A_{t},$

$\begin{equation} E_{x} = \frac{-1}{2}C_{t},\text{ }E_{\Phi} = \frac{-1}{4}A_{t},\text{ }E_{y} = \frac{-1}{2}B_{t}. \end{equation}$

(7)

solving the system of Eq (7), we get

$\begin{align*} A & = g_{1}(t),\text{ }B = \frac{1}{4}g_{1}^{\prime}x+g_{3}(t),\text{ }C = \frac{1}{2}g_{1}^{\prime}y+g_{2}(t),\text{ }\nonumber\\ E & = \frac{-1}{4}g_{1}^{\prime}\Phi-\frac{1}{4}xyg_{1}^{\prime\prime} -yg_{2}^{\prime}-\frac{1}{2}xg_{3}^{\prime}+g_{4}(t). \end{align*}$

(8)

The vector field operator $V$ of Eq (8) can be presented as

$\begin{equation} V = V_{1}(c_{1})+V_{2}(c_{2})+V_{3}(c_{3})+V_{4}(c_{4}), \end{equation}$

(9)

where

$\begin{align*} V_{1} & = g_{1}(t)\left( \frac{\partial}{\partial t}+\frac{1}{4} x\frac{\partial}{\partial x}+\frac{1}{2}y\frac{\partial}{\partial y}-\frac {1}{4}\Phi\frac{\partial}{\partial\Phi}\right) -\frac{1}{4}xyg_{1} ^{\prime\prime}\frac{\partial}{\partial\Phi},\nonumber\\ V_{2} & = g_{2}(t)\frac{\partial}{\partial y}-y\alpha_{2}^{\prime} \frac{\partial}{\partial\Phi},\text{ }V_{3} = g_{3}(t)\frac{\partial }{\partial x}-\frac{1}{2}g_{3}^{\prime}\frac{\partial}{\partial\Phi},\text{ }V_{4} = g_{4}(t)\frac{\partial}{\partial\Phi}. \end{align*}$

(10)

The Lie-bracket $[V_{i}, V_{j}] = V_{i}V_{j}-V_{j}V_{i}$ is used for obtaining the commutator relations which are shown in Table 1, where

$\begin{align*} l_{1} & = g_{1}g_{2}^{\prime}-\frac{1}{2}g_{2}g_{1}^{\prime},\text{ } l_{2} = \frac{-1}{4}(g_{2}^{\prime}g_{1}^{\prime}+2g_{1}g_{2}^{\prime\prime }-g_{2}g_{1}^{\prime\prime})x,\nonumber\\ l_{3} & = g_{1}g_{3}^{\prime}-\frac{1}{4}g_{3}g_{1}^{\prime},\text{ } l_{4} = \frac{-1}{4}(g_{3}^{\prime}g_{1}^{\prime}+4g_{1}g_{3}^{\prime\prime }-g_{3}g_{1}^{\prime\prime})y,\nonumber\\ l_{5} & = \frac{1}{4}(4g_{1}g_{4}^{\prime}+g_{4}g_{1}^{\prime}),\text{ } l_{6} = \frac{-1}{2}(2g_{2}g_{3}^{\prime}-g_{3}g_{2}^{\prime}). \end{align*}$

(11)

Table 1. The commutator table.

	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4}$
$V_{1}$	$0$	$l_{1}\frac{\partial}{\partial y}+l_{2}\frac{\partial }{\partial\Phi}$	$l_{3}\frac{\partial}{\partial x}+a_{4}\frac{\partial }{\partial\Phi}$	$l_{5}\frac{\partial}{\partial\Phi}$
$V_{2}$	-( $l_{1}\frac{\partial}{\partial y}+l_{2}\frac{\partial} {\partial\Phi})$	$0$	$l_{6}\frac{\partial}{\partial\Phi}$	$0$
$V_{3}$	-( $l_{3}\frac{\partial}{\partial x}+l_{4}\frac{\partial} {\partial\Phi})$	- $l_{6}\frac{\partial}{\partial\Phi}$	$0$	$0$
$V_{4}$	$-l_{5}\frac{\partial}{\partial\Phi}$	- $l_{6}\frac{\partial }{\partial\Phi}$	$0$	$0$

| Show Table

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3. Reductions and closed form solutions

To obtain new independent variables, we utilize the characteristic equation as follows:

$\begin{equation} \frac{dt}{A(t,x,y,\Phi)} = \frac{dx}{B(t,x,y,\Phi)} = \frac{dy}{C(t,x,y,\Phi )} = \frac{d\Phi}{E(t,x,y,\Phi)}. \end{equation}$

(12)

Case I. Using $V_{1}$ where $g_{1}(t) = t^{2},$ Eq (12) takes the following form:

$\begin{equation} \frac{dt}{t^{2}} = \frac{dx}{\frac{1}{2}tx} = \frac{dy}{ty} = \frac{d\Phi}{\frac {-1}{2}t\Phi}. \end{equation}$

(13)

The similarity solution of Eq (1) takes the form

$\begin{equation} \Phi(t,x,y) = \frac{1}{\sqrt{t}}F(\zeta,\eta)-\frac{xy}{2t}, \end{equation}$

(14)

where $\zeta = \frac{x}{\sqrt{t}}, \; \eta = \frac{y}{t}.$ Substituting $\Phi$ in Eq (1) we get

$\begin{equation} F_{\eta\eta\eta}+6F_{\zeta\zeta}F_{\zeta\eta}+2F_{\zeta\zeta\zeta }F_{\eta}+4F_{\zeta\zeta\eta}F_{\zeta}+F_{\zeta\zeta\zeta\zeta\eta } = 0. \end{equation}$

(15)

The ordinary differential equations of Eq (15) take the following experssion:

$\begin{equation} h^{3}F^{\prime\prime\prime}(\theta)+6k^{3}hF^{\prime\prime^{2}}(\theta )+6k^{3}hF^{\prime\prime\prime}(\theta)F^{\prime}(\theta)+k^{4}hF^{\prime \prime\prime\prime\prime}(\theta) = 0, \end{equation}$

(16)

where $\theta = k\zeta+h\eta.$

To obtain the exact solution for Eq (16), we utilize generalized Kudryashov method (GKM) ^[11], given as follows:

$\begin{equation} F(\theta) = A_{0}+\sum\limits_{i=1}^m\frac{A_{i}}{(1+\phi (\theta))^{i}}, \end{equation}$

(17)

where $\phi(\theta)$ is the solution for the following Ricati equation $\phi^{\prime}(\theta) = A+B\phi(\theta)+C\phi^{2}(\theta).$ For simplicity, we take $i = 1$ and substitute Eq (17) into Eq (16) then to obtain

$\begin{equation} h = \pm\sqrt{4AC-B^{2}}k^{2},\text{ }A_{1} = 2k(C-B+A),\text{ }k\text{ and }A_{0}\text{ are arbitrary constant.} \end{equation}$

(18)

Substituting Eq (18) into Eqs (14) and (17), the exact solutions of Eq (1) can be expressed as

Set 1. $A\neq0, B\neq0, C\neq0.$

$\begin{equation} \Phi(t,x,y) = \frac{1}{\sqrt{t}}\left( A_{0}+\frac{2k(C-B+A)}{{\Large [} \frac{-B}{2C}\text{+}\frac{\sqrt{4AC-B^{2}}}{2C}\tan\text{(}\frac{1} {2}\text{(}\sqrt{4AC-B^{2}}\theta\text{))}+1{\LARGE ]}}\right) -\frac{xy} {2t}, \end{equation}$

(19)

where $\theta = k\frac{x}{\sqrt{t}}+\sqrt{4AC-B^{2}}k^{2}\frac{y}{t}.$

Set 2. $A = \frac{-1}{2}, B = 0, C = \frac{-1}{2}.$

$\begin{equation} \Phi(t,x,y) = \frac{1}{\sqrt{t}}\left( A_{0}+\frac{-2k}{\cot(\theta)\pm \csc(\theta)+1}\right) -\frac{xy}{2t}, \end{equation}$

(20)

where $\theta = k\frac{x}{\sqrt{t}}+2k^{2}\frac{y}{t}.$

Case II. Substituting $V_{1}$ in Eq (12) where $g_{1}(t) = t,$ then Eq (12) is expressed as follows:

$\begin{equation} \frac{dt}{t} = \frac{dx}{\frac{1}{4}x} = \frac{dy}{\frac{1}{2}y} = \frac{d\Phi }{\frac{-1}{4}\Phi}. \end{equation}$

(21)

Then the invariants are presented by $\zeta = \frac{x}{t^{\frac{1}{4}}}, \; \eta = \frac{y}{t^{\frac{1}{2}}}$ and $\Phi(t, x, y) = \frac{1}{t^{\frac{1}{4}} }F(\zeta, \eta).$ Further Eq (1) is reduced to the following form:

$\begin{align*} \frac{3}{2}F_{\zeta\zeta}+\frac{1}{2}\zeta F_{\zeta\zeta\zeta}+\eta F_{\eta\eta\eta}+F_{\eta\eta\eta}+6F_{\zeta\zeta}F_{\zeta\eta}+ 4F_{\zeta}F_{\zeta\zeta\eta}+2F_{\zeta\zeta\zeta}F_{\eta}+F_{\zeta \zeta\zeta\zeta\eta} = 0. \end{align*}$

(22)

For reducing Eq (22) to ODE we have two subcases:

Subcase (a). Using the scaling transformation in Eq (22)

$\begin{equation} F = e^{\epsilon}\text{ }\overset{-}{F},\text{ }\zeta = e^{\epsilon_{1}}\text{ }\overset{-}{\zeta},\text{ }\eta = e^{\epsilon_{2}}\text{ }\eta. \end{equation}$

(23)

Substituting from Eq (23) into Eq (22) we have $2\epsilon_{1} = \epsilon _{2} = -2\epsilon.$

The characteristic is given as

$\begin{equation} \frac{2d\zeta}{\zeta} = \frac{d\eta}{\eta} = \frac{-2dF}{F}. \end{equation}$

(24)

The invariant variables can be written as

$\begin{equation} \theta = \frac{\zeta^{2}}{\eta},\text{ }f = \frac{1}{\zeta^{2}}F(\theta). \end{equation}$

(25)

The ODE of Eq (22) is provided by

$\begin{align*} -\theta^{2}f^{\prime\prime\prime}-6f^{\prime\prime}-6\theta f^{\prime }-48\theta f^{\prime\prime^{2}}-48\theta ff^{\prime}-48f^{\prime} f^{\prime\prime\prime}-16f^{\prime}f^{\prime\prime\prime}-80\theta f^{\prime\prime\prime} -60f^{\prime\prime\prime}-16\theta^{2}f^{\prime\prime\prime\prime\prime} = 0. \end{align*}$

(26)

To obtain the exact solutions of Eq (26), we suppose that the solution of Eq (26) has the form

$\begin{equation} f(\theta) = A_{0}+\sum\limits_{j=1}^nA_{j}\theta^{j}. \end{equation}$

(27)

For simplicity, we take $j = 2$ , substitute Eq (27) into Eq (26), then the closed form solution of Eq (1) takes the expression

$\begin{equation} \Phi(x,y) = A_{0}+A_{1}\frac{x}{y}-\frac{1}{16}\frac{x^{3}}{y^{2}}, \end{equation}$

(28)

where $A_{0}$ and $A_{1}$ are arbitrary constant.

Subcase (b). Utilizing symmetry method for Eq (22), we obtained the infinitesimals of Eq (22) as follows:

$\begin{equation} \tau = \frac{1}{2}c_{1}\zeta+c_{2},\text{ }\beta = c_{1}\eta+c_{3},\text{ }\psi = \frac{-1}{2}(F+xy)c_{1}-\frac{1}{4}\eta c_{2}-\frac{1}{4}\zeta c_{3}. \end{equation}$

(29)

The vector field of Eq (22) can take the form

$\begin{equation} \chi = \chi_{1}(c_{1})+\chi_{2}(c_{2})+\chi_{3}(c_{3}), \end{equation}$

(30)

where

$\begin{equation} \chi_{1} = \frac{1}{2}\zeta\frac{\partial}{\partial\zeta}+\frac{\partial }{\partial\eta}-\frac{1}{2}(F+xy)\frac{\partial}{\partial F},\text{ }\chi _{2} = \frac{\partial}{\partial\zeta}-\frac{1}{4}\eta\frac{\partial}{\partial F},\text{ }\chi_{3} = \frac{\partial}{\partial\eta}-\frac{1}{4}\zeta \frac{\partial}{\partial F}. \end{equation}$

(31)

Using $\chi_{2}+w\chi_{3}$ , the characteristic equation is yielded

$\begin{equation} \frac{d\zeta}{1} = \frac{d\eta}{w} = \frac{dF}{-\frac{1}{4}\eta-\frac{w}{4}\zeta }. \end{equation}$

(32)

The similarity solution for Eq (22) takes the form

$\begin{equation} F(\zeta,\eta) = f(\theta)-\frac{1}{4}\eta\zeta,\text{ where }\theta = \eta-w\zeta. \end{equation}$

(33)

The reduced ODE can take the expression

$\begin{equation} f^{\prime\prime\prime}(\theta)-6w^{3}f^{\prime\prime2}(\theta)-6w^{3} f^{\prime}(\theta)f^{\prime\prime\prime}(\theta)+w^{4}f^{\prime\prime \prime\prime\prime}(\theta) = 0. \end{equation}$

(34)

Utilizing GKM in the previous case, the closed-form solution of Eq (34) is shown as

$\begin{equation} f(\theta) = A_{0}+\frac{A_{1}}{1+\phi(\theta)}+\frac{A_{2}}{(1+\phi(\theta ))^{2}}. \end{equation}$

(35)

Substituting Eq (35) into Eq (34), collecting the coefficients of derivative $\phi(\theta)$ and solving the algebraic equations, we obtain the values of $A_{0}$ and $A_{1}$

$\begin{equation} A_{1} = \frac{-2(A+C-B)}{(4AC-B^{2})^{\frac{1}{4}}},\text{ }w = \frac {1}{(4AC-B^{2})^{\frac{1}{4}}},\text{ }A_{0},\text{ }A_{2}\text{ are arbitrary.} \end{equation}$

(36)

Set 1. $A\neq0, B\neq0, C\neq0.$

$\begin{align*} \Phi(t,x,y) & = \frac{1}{t^{\frac{1}{4}}}\left( A_{0}+\frac{A_{1} }{{\Large [}\frac{-B}{2C}\text{+}\frac{\sqrt{4AC-B^{2}}}{2C}\tan\text{(} \frac{1}{2}\text{(}\sqrt{4AC-B^{2}}\theta\text{))}+1{\LARGE ]}}\right. \nonumber\\ & \left. +\frac{A_{2}}{{\Large [}\frac{-B}{2C}\text{+}\frac{\sqrt{4AC-B^{2} }}{2C}\tan\text{(}\frac{1}{2}\text{(}\sqrt{4AC-B^{2}}\theta\text{))} +1{\LARGE ]}^{2}}\right) -\frac{y}{4\sqrt{t}}\frac{x}{t^{\frac{1}{4}}}, \end{align*}$

(37)

where $A_{1} = \frac{-2(A+C-B)}{(4AC-B^{2})^{\frac{1}{4}}}, \; \theta = \frac {y}{\sqrt{t}}+\frac{1}{(4AC-B^{2})^{\frac{1}{4}}}\frac{x}{t^{\frac{1}{4}}}.$

Set 2. $A = \frac{1}{2}, B = 0, C = \frac{1}{2}.$

$\begin{equation} \Phi(t,x,y) = \frac{1}{t^{\frac{1}{4}}}\left( A_{0}+\frac{A_{1}}{\csc (\theta)-\cot(\theta)+1}+\frac{A_{2}}{(\csc(\theta)-\cot(\theta)+1)^{2} }\right) -\frac{y}{4\sqrt{t}}\frac{x}{t^{\frac{1}{4}}}, \end{equation}$

(38)

where $A_{1} = \frac{-1}{2}, \; \theta = \frac{y}{\sqrt{t}}+\frac{x}{t^{\frac{1} {4}}}.$

Cases III. Using $\mathbf{w}V_{2}, V_{3}, V_{4}$ where $g_{2}(t), \; g_{3}(t), \; g_{4}(t)$ are constants, we get

$\begin{equation} \frac{dt}{0} = \frac{dx}{w} = \frac{dy}{1} = \frac{d\Psi}{1}. \end{equation}$

(39)

Using Eq (39), we obtain the invariant solutions

$\begin{equation} \Phi(t,x,y) = F(\zeta,\eta)+y,\text{ where }\zeta = t,\text{ }\eta = x-wy. \end{equation}$

(40)

Substituting Eq (40) into Eq (1) we get

$\begin{equation} 2F_{\zeta\eta\eta}+w^{3}F_{\eta\eta\eta}+6wF_{\eta\eta}^{2}+6wF_{\eta \eta\eta}F_{\eta}-2F_{\eta\eta\eta}+wF_{\eta\eta\eta\eta\eta } = 0. \end{equation}$

(41)

Utilizing transformation $\theta = k\zeta+h\eta$ , Eq (41) is reduced to the next equation

$\begin{equation} h^{2}(2k+hw^{3}-2h)F^{\prime\prime\prime}(\theta)+6wh^{4}F^{\prime\prime^{2} }(\theta)+6wh^{4}F^{\prime\prime\prime}(\theta)F^{\prime}(\theta )+wh^{5}F^{\prime\prime\prime\prime\prime}(\theta) = 0. \end{equation}$

(42)

The exact solutions of Eq (42) can be deduced by applying GKM in the previous method. Substituting Eq (17) into Eq (42) at $i = 1$ and collecting the coefficient of derivative $\phi(\theta),$ then equating this equations to zero, this equations is solved and the values of $A_{0}, \; A_{1}, w, h$ and $k$ are obtained as

$\begin{align*} k = \frac{1}{2}(4h^{2}wAC+4-w^{3}-h^{2}wB^{2}),\text{ }A_{1} = 2(C+A-B)h,\; \; A_{0},h,w\text{ are arbitrary.} \end{align*}$

(43)

The closed-form solutions of Eq (1) can be expressed as

Set 1. $A\neq0, B\neq0, C = 0.$

$\begin{equation} \Phi(t,x,y) = A_{0}+\frac{2(A-B)h}{\frac{-A}{B}\text{+}\frac{1}{B}\exp \text{(}B\text{ }\theta\text{)}+1}+y, \end{equation}$

(44)

where $k = \frac{1}{2}(4-w^{3}-h^{2}wB^{2}), \; \theta = kt+h(x-wy).$

Set 2. $A = 0, B\neq0, C\neq0.$

$\begin{equation} \Phi(t,x,y) = A_{0}+\frac{2(C-B)h\text{ }(C\exp\text{(}B\text{ }\theta \text{)}-1)}{B\exp\text{(}B\text{ }\theta\text{)}+C\exp\text{(}B\text{ } \theta\text{)}-1}+y, \end{equation}$

(45)

where $k = \frac{1}{2}(4-w^{3}-h^{2}wB^{2}), \; \theta = kt+h(x-wy).$

Cases IV. Utilizing $wV_{2}, V_{3},$ where $\alpha _{2}(t), \alpha_{3}(t)$ are constant, we obtain

$\begin{equation} \frac{dt}{0} = \frac{dx}{1} = \frac{dy}{w} = \frac{d\Psi}{0}. \end{equation}$

(46)

Using Eq (46), we obtain the invariant solutions

$\begin{equation} \Phi(t,x,y) = F(\zeta,\eta),\text{ where }\zeta = t,\text{ }\eta = x-wy. \end{equation}$

(47)

Substituting Eq (47) into Eq (1) we get

$\begin{equation} 2F_{\zeta\eta\eta}+w^{3}F_{\eta\eta\eta}++6wF_{\eta\eta}^{2}+6wF_{\eta \eta\eta}F_{\eta}+wF_{\eta\eta\eta\eta\eta} = 0. \end{equation}$

(48)

Taking the transformation $\theta = k\zeta+h\eta$ , Eq (48) is reduced to the form

(49)

The coveted exact solution of Eq (49) is given. Hence, by using Eq (17) in Eq (49), the solution for Eq (1) is provided as

Set 1. $A\neq0, B\neq0, C\neq0.$

$\begin{equation} \Phi(t,x,y) = A_{0}+\frac{2(C+A-B)h}{{\Large [}\frac{-B}{2C}+\frac {\sqrt{4AC-B^{2}}}{2C}\tan(\frac{1}{2}(\sqrt{4AC-B^{2}}\theta))+1{\LARGE ]}}, \end{equation}$

(50)

where $k = \frac{1}{2}(4h^{2}wAC+4-w^{3}-h^{2}wB^{2}), \; \theta = kt+h(x-wy).$

Set 2. $A = \frac{1}{2}, B = 0, C = \frac{-1}{2}.$

$\begin{equation} \Phi(t,x,y) = A_{0}-\frac{1}{2}\frac{h}{\coth(\theta)\pm\csc h(\theta)+1}, \end{equation}$

(51)

where $k = \frac{1}{2}(-h^{2}w+4-w^{3}), \; \theta = kt+h(x-wy).$

4. Results & discussion

In this section, we explain the dynamic behavior of the obtained solutions in Eqs (19), (20), (28), (37), (38), (44), (45) and compare these results with previous literatures.

The physical behavior of the solutions is illustrated in three-dimensional and two-dimensional forms. The solutions include trigonometric functions, hyperbolic functions, and expansion functions. The Maple software was used to review the solutions as follows:

In , presents wave solution of Eq (19) in 3-D and 2-D shapes with perfect numerical values of $A_{0} = 5, \; k = 0.1, \; A = 3, \; C = 2, \; B = -1$ at y = 1.

Figure 1. Wave solution of Eq (19) in 3-D and 2-D shapes.

DownLoad: Full-Size Img PowerPoint

In , the solution of Eq (28) shows the behavior of a single wave and it is presented through 3-D and 2-D graphs. The constants are chosen as follows: $A_{0} = 1, A_{2} = 1, \; k = 0.1, \; A = 3, \; C = 2, \; B = 3$ when $y = 1$ .

Figure 2. Single wave of Eq (28) in 3-D and 2-D graphs.

DownLoad: Full-Size Img PowerPoint

shows doubly soliton of Eq (37) in 3-D and 2-D shapes with perfect choice for values of $A_{0} = 0, \; A_{1} = 1$ .

Figure 3. Doubly soliton of Eq (37) in 3-D and 2-D shapes.

DownLoad: Full-Size Img PowerPoint

depicts the view anti-kink wave solution of Eq (44) in 3-D and 2-D graphs with a suitable choice of values $A_{0} = 3, w = -1, h = -3, A = 2, C = 0, B = 1$ with $y = 1.$

Figure 4. Anti-kink wave solution of Eq (44) in 3-D and 2-D graphs.

DownLoad: Full-Size Img PowerPoint

In , the solution of Eq (45) represents a kink wave solution in 3-D and 2-D plots with perfect simulation of the values $A_{0} \; = 1, \; h = 2, w = 1, \; z = 1, A = 0, \; C = -3, B = 1$ at $y = 1.$

Figure 5. Kink wave solution of Eq (45) in 3-D and 2-D plots.

DownLoad: Full-Size Img PowerPoint

The authors in ^[13] used the symmetry method and obtained a subcase from our infinitesimals. Then, they reduced the governing equation to one reduction case, but we obtained five reduction cases.

The authors in ^[14] used two different traveling wave methods to obtain exact solutions. However, they reduced the governing equation through an algebraic transformation. When comparing these solutions with our obtained solutions, it turned out that our solutions are novel solutions and show different physical behavior from other solutions.

5. Conclusions

In this paper, we applied the Lie symmetry method to obtain infinitesimals and vector fields. This led to reduce the governing equation to a new form of partial differential equations (PDEs). Then, by utilizing the symmetry method again and similarity transformation: the new form of PDEs has been converted instead of reduced to five different ODEs. The general exact wave solutions are obtained by using GKM. These solutions are represented in 2-D and 3-D graphs to show the physical properties of the solutions.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

The author (A. A. Gaber) would like to thank the Deanship of Scientific Research of Majmaah University for supporting this work under Project Number (R-2024-947).

Conflict of interest

The authors declare no conflicts of interest.

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AIMS Mathematics

A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations

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Abstract

1. Introduction

1.1. Overview

1.2. Related literatures

1.3. Main idea

2. Lie symmetry method

3. Reductions and closed form solutions

4. Results & discussion

5. Conclusions

Use of AI tools declaration

Acknowledgment

Conflict of interest

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1.1. Overview

1.2. Related literatures

1.3. Main idea

2. Lie symmetry method

3. Reductions and closed form solutions

4. Results & discussion

5. Conclusions

Use of AI tools declaration

Acknowledgment

Conflict of interest

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AIMS Mathematics

A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations

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Abstract

1. Introduction

1.1. Overview

1.2. Related literatures

1.3. Main idea

2. Lie symmetry method

3. Reductions and closed form solutions

4. Results & discussion

5. Conclusions

Use of AI tools declaration

Acknowledgment

Conflict of interest

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Abstract

1. Introduction

1.1. Overview

1.2. Related literatures

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2. Lie symmetry method

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4. Results & discussion

5. Conclusions

Use of AI tools declaration

Acknowledgment

Conflict of interest

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