# Computational and Numerical Solutions for2+1-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform

Authors: Raghda A. M. Attia; S. H. Alfalqi; J. F. Alzaidi; Mostafa M. A. Khater; Dianchen Lu
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the analytical, semianalytical, and numerical solutions of the2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.


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## Body

## 1. Introduction

The Korteweg–de Vries (KdV) equation is a seminal model in fluid mechanics. This model was introduced by Boussinesq in 1877 and reintroduced by Diederik Korteweg and Gustav de Vries in 1895. The KdV has the following formula [1–9]:(1)Qt+Uxxx−6UUx=0,where U=Ux,t characterizes the weakly nonlinear shallow water waves. Equation (1) can be written in many distinct forms and combined with other models. One of them is the Schwarz–Korteweg–de Vries (SKdV) equation given by(2)Qt+QxQxxQx−12QxxQx2=0,where Q=Qx,t is the unknown function. The SKdV was derived by Krichever and Novikov [10] and Weiss [11, 12].In this paper, we study the2+1-dimensional integrable generalization of SKdV as follows:(3)Ut+14Uxxy−UxUxy2U−UxxUy4U+Ux2Uy2U2−Ux8∫Ux2U2ydx=0.Equation (3) was derived by Toda and Yu [13]. Using the following transformation on equation (3),(4)U=Sx,S=eF,Fx=U,Ft=R,where S=Sx,t,F=Fx,t,U=Ux,t, and R=Rx,t are the unknown functions, we obtain(5)4U2Rx−4UUxR+U2Uxxy−UUxxUy−3UUxUxy+3Ux2Uy−U4Uy=0,Ut−Rx=0.This equation is obtained from the study by Bruzón et al. [14–16]. Using the Miura transform [17–19] on equation (5) as(6)Bx=Uxx4U−3Ux28U2−U28,By=−RU,we obtain [20, 21](7)4Bxt+Bxxxy+8BxyBx+4ByBxx=0.If we adopt the wave transformation(8)Bx,y,t=Bℨ,ℨ=ρx+δy+ct,then we convert equation (7) into an ordinary differential equation (NLODE). The integration of the obtained NLODE with zero constant of integration leads to(9)4cB′+ρ2δB‴+6ρδB′2=0.If we consider the substitutionB′=F, then it results in(10)4cF+ρ2δF″+6ρδF2=0.Having these ideas in mind, this paper is organized as follows: Section2 presents the two methods and derives the solutions of the SKdV equation. Section 5 represents the solutions for several numerical values of the parameters. Additionally, their stability and properties are also discussed. Finally, Section 6 summarises the main conclusions.


## 2. Explicit Solutions

In this section, we apply two analytical techniques for deriving the solutions of the2+1-dimensional integrable SKdV model. We adopt the extended simplest equation method and the sech-tanh method to obtain various distinct formulas of solitary wave solutions of equation (3). For further details about the two methods, see [22–26].


### 2.1. Extended Simplest Equation

According to the homogeneous balance rule between the highest derivative and the nonlinear term in equation (9), we obtain n=2. Thus, the general solution of equation (10) is given by(11)Fℨ=∑i=−nnaiCℨi=a2Cℨ2+a1Cℨ+a−2Cℨ2+a−1Cℨ+a0,where aii=−2,…,2 are arbitrary constants. Additionally, Cℨ satisfies the following ordinary differential equation:(12)C′ℨ=α+λCℨ+μCℨ2,where β,α,andμ are the arbitrary constants. Substituting equations (11) and (12) into (9) and collecting all terms of Ciℨi=−4,−3,…,3,4, we get a system of algebraic equations. Solving this system, we obtain two families of solutions.Family 1.
(13)a0⟶16−2αμρ−λ2ρ,a1⟶0,a2⟶0,a−1⟶−αλρ,a−2⟶−α2ρ,c⟶14δλ2ρ2−4αδμρ2.Family 2.
(14)a0⟶16−2αμρ−λ2ρ,a1⟶−λμρ,a2⟶−μ2ρ,a−1⟶0,a−2⟶0,c⟶14δλ2ρ2−4αδμρ2.
From these two families, the solitary wave solutions of equation (7) can be obtained.
According toFamily 1, we have the following expressions.


#### 2.1.1. Whenλ=0

Forαμ>0, we obtain(15)F1x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(16)F2x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.Forαμ<0, we obtain(17)F3x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(18)F4x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.When4αμ>λ2, we obtain(19)F5x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ,(20)F6x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ.According toFamily 2, we have the following expressions.


#### 2.1.2. Whenλ=0

Forαμ>0, we obtain(21)F7x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(22)F8x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.Forαμ<0, we obtain(23)F9x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(24)F10x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.Whenα=0: For λ>0, we get(25)F11x,y,t=−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−2λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12.Forλ<0, we obtain(26)F12x,y,t=−μ4ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ2ρe1/4λρδλ2ρt+4x+δλy+λϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ3ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12.When4αμ>λ2, we obtain(27)F13x,y,t=2αμρ3−λ2ρ6+14λ2ρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ,(28)F14x,y,t=2αμρ3−λ2ρ6+14λ2ρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ.


### 2.2. Sech-Tanh Method

The general solution of equation (10) according to the sech-tanh method and calculated value of balance is given by(29)Fℨ=∑i=1nsechi−1ℨaisechℨ+bitanhℨ+a0=sechℨa2sechℨ+b2tanhℨ+a1sechℨ+a0+b1tanhℨ,where a0,a1,a2,b1,andb2 are the arbitrary constants. Substituting equation (29) into (10) and collecting all terms of sechℨ,sech2ℨ,sech3ℨ,tanhℨ,tanhℨsech2ℨ,andtanhξsechℨ, we obtain a system of algebraic equations. Solving this system, we obtain(30)a0⟶2c3δ,a1⟶0,a2⟶−cδ,b1⟶0,b2⟶0,ρ⟶−cδ,wherec>0,δ>0.Consequently, the explicit solution of equation (7) is given by(31)F15x,y,t=c2−3sech2ct+ρx+δy3δ.


## 2.1. Extended Simplest Equation

According to the homogeneous balance rule between the highest derivative and the nonlinear term in equation (9), we obtain n=2. Thus, the general solution of equation (10) is given by(11)Fℨ=∑i=−nnaiCℨi=a2Cℨ2+a1Cℨ+a−2Cℨ2+a−1Cℨ+a0,where aii=−2,…,2 are arbitrary constants. Additionally, Cℨ satisfies the following ordinary differential equation:(12)C′ℨ=α+λCℨ+μCℨ2,where β,α,andμ are the arbitrary constants. Substituting equations (11) and (12) into (9) and collecting all terms of Ciℨi=−4,−3,…,3,4, we get a system of algebraic equations. Solving this system, we obtain two families of solutions.Family 1.
(13)a0⟶16−2αμρ−λ2ρ,a1⟶0,a2⟶0,a−1⟶−αλρ,a−2⟶−α2ρ,c⟶14δλ2ρ2−4αδμρ2.Family 2.
(14)a0⟶16−2αμρ−λ2ρ,a1⟶−λμρ,a2⟶−μ2ρ,a−1⟶0,a−2⟶0,c⟶14δλ2ρ2−4αδμρ2.
From these two families, the solitary wave solutions of equation (7) can be obtained.
According toFamily 1, we have the following expressions.


### 2.1.1. Whenλ=0

Forαμ>0, we obtain(15)F1x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(16)F2x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.Forαμ<0, we obtain(17)F3x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(18)F4x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.When4αμ>λ2, we obtain(19)F5x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ,(20)F6x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ.According toFamily 2, we have the following expressions.


### 2.1.2. Whenλ=0

Forαμ>0, we obtain(21)F7x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(22)F8x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.Forαμ<0, we obtain(23)F9x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(24)F10x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.Whenα=0: For λ>0, we get(25)F11x,y,t=−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−2λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12.Forλ<0, we obtain(26)F12x,y,t=−μ4ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ2ρe1/4λρδλ2ρt+4x+δλy+λϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ3ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12.When4αμ>λ2, we obtain(27)F13x,y,t=2αμρ3−λ2ρ6+14λ2ρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ,(28)F14x,y,t=2αμρ3−λ2ρ6+14λ2ρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ.


## 2.1.1. Whenλ=0

Forαμ>0, we obtain(15)F1x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(16)F2x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.Forαμ<0, we obtain(17)F3x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(18)F4x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.When4αμ>λ2, we obtain(19)F5x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2tan1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ,(20)F6x,y,t=−αμρ3−4α2μ2ρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ2−λ2ρ6+2αλμρλ−4αμ−λ2cot1/84αμ−λ2ρδρtλ2−4αμ+4x+4δy+4ϑ.According toFamily 2, we have the following expressions.


## 2.1.2. Whenλ=0

Forαμ>0, we obtain(21)F7x,y,t=−αμρ3−λ2ρ6−αμρtan214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμtan14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ,(22)F8x,y,t=−αμρ3−λ2ρ6−αμρcot214αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ−λραμcot14αμδρ2tλ2−4αμ+4ρx+4δy+4ϑ.Forαμ<0, we obtain(23)F9x,y,t=αμρtanh214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3,(24)F10x,y,t=αμρcoth214−αμρ4x−4αδμρt+4δy∓logϑ2−αμρ3.Whenα=0: For λ>0, we get(25)F11x,y,t=−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−2λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ−12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ−12.Forλ<0, we obtain(26)F12x,y,t=−μ4ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2ρ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μ2ρexp1/2λρδλ2ρt+4x+4δy+4ϑ6μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ2ρe1/4λρδλ2ρt+4x+δλy+λϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12−λ2μρe1/4λρδλ2ρt+4x+δλy+λϑ3μe1/4δλ3ρ2t+λρx+δλy+λϑ+12+λμ3ρexp1/2λρδλ2ρt+4x+4δy+4ϑμe1/4δλ3ρ2t+λρx+δλy+λϑ+12.When4αμ>λ2, we obtain(27)F13x,y,t=2αμρ3−λ2ρ6+14λ2ρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρsec2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ,(28)F14x,y,t=2αμρ3−λ2ρ6+14λ2ρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ−αμρcsc2184αμ−λ2δρ2tλ2−4αμ+4ρx+4δy+4ϑ.


## 2.2. Sech-Tanh Method

The general solution of equation (10) according to the sech-tanh method and calculated value of balance is given by(29)Fℨ=∑i=1nsechi−1ℨaisechℨ+bitanhℨ+a0=sechℨa2sechℨ+b2tanhℨ+a1sechℨ+a0+b1tanhℨ,where a0,a1,a2,b1,andb2 are the arbitrary constants. Substituting equation (29) into (10) and collecting all terms of sechℨ,sech2ℨ,sech3ℨ,tanhℨ,tanhℨsech2ℨ,andtanhξsechℨ, we obtain a system of algebraic equations. Solving this system, we obtain(30)a0⟶2c3δ,a1⟶0,a2⟶−cδ,b1⟶0,b2⟶0,ρ⟶−cδ,wherec>0,δ>0.Consequently, the explicit solution of equation (7) is given by(31)F15x,y,t=c2−3sech2ct+ρx+δy3δ.


## 3. Stability Investigation

We now examine the stability property for2+1-dimensional integrable SKdV model with the Miura transformation by means of an Hamiltonian system. The momentum ℌ in the Hamiltonian system is given by(32)ℌ=12∫−JJB2ℨdℨ,where Bℨ is the solution of the model. Consequently, the condition for stability of the solutions can be formulated as(33)∂ℌ∂c>0,where c is the wave velocity. The momentum in the Hamiltonian system for equations (18) and (31) are given, respectively, by(34)ℌ=1c3200c−sech210c+26−sech210c+34+sech226−10c+sech234−10c−4loge52−1×sinh10c+1+e52cosh10c−4loge68−1sinh10c+1+e68cosh10c+4log1+e52cosh10c−e52−1sinh10c+4log1+e68cosh10c−e68−1sinh10c,(35)ℌ=1724100c+loge7−5c+e5c+loge23−5c+e5c−loge−5c+e5c+7−loge−5c+e5c+23−sech25c+72−sech25c+232+sech272−5c+sech2232−5c.And thus,(36)∂ℌ∂cc=72=2.37146×10−298>0,(37)∂ℌ∂cc=9=5.55556>0.We conclude that this solution is stable on the intervalx∈−5,5,t∈−5,5. This result shows the ability of the solutions for their application. Using the same steps, we can check the stability property of all other obtained solutions.


## 4. Semianalytical and Numerical Solutions

This section applies semianalytical and numerical schemes for deriving the solutions of the2+1-dimensional integrable SKdV model. The Adomian decomposition method and cubic b-spline schemes are employed to the method to investigate the accuracy of the obtained analytical solutions. Also, this study aims to give a comparison between both used analytical schemes. For further details about the two methods, see [27–30].


### 4.1. Adomian Decomposition Method

Applying this scheme gives equation (10) in the following form:(38)∑j=0∞Fjℨ=F0+F′0ℨ−4cρ2δL−1∑j=0∞Fjℨ−6ρL−1∑j=0∞Aj.Thus, with respect to equation (18) and the following conditions α=−1,a0=4,a−1=0,a−2=−3,c=72,δ=2,λ=0,μ=4,andρ=3, we obtain(39)F0=−2,F1=12ℨ2,F2=−8ℨ4,F3=8ℨ4−16ℨ63.Consequently, the semianalytical solution of equation (10) is written in the following form:(40)F=−16ℨ63+12ℨ2−2+⋯.However, with respect to equation (31) and the following conditions a0=1,a1=0,a2=−3/2,b1=0,b2=0,c=9,δ=4,andρ=−3/2, we obtain(41)F0=−12,F1=3ℨ22,F2=−ℨ4,F3=13ℨ630−ℨ42.Consequently, the semianalytical solution of equation (10) is written in the following form:(42)F=13ℨ630−3ℨ42+3ℨ22−12+⋯.


### 4.2. Cubic B-Spline Scheme

Employing the cubic B-spline scheme to evaluate the numerical solutions of equation (10). Using same initial and boundary conditions with respect to the obtained solutions (18) and (31), yields


## 4.1. Adomian Decomposition Method

Applying this scheme gives equation (10) in the following form:(38)∑j=0∞Fjℨ=F0+F′0ℨ−4cρ2δL−1∑j=0∞Fjℨ−6ρL−1∑j=0∞Aj.Thus, with respect to equation (18) and the following conditions α=−1,a0=4,a−1=0,a−2=−3,c=72,δ=2,λ=0,μ=4,andρ=3, we obtain(39)F0=−2,F1=12ℨ2,F2=−8ℨ4,F3=8ℨ4−16ℨ63.Consequently, the semianalytical solution of equation (10) is written in the following form:(40)F=−16ℨ63+12ℨ2−2+⋯.However, with respect to equation (31) and the following conditions a0=1,a1=0,a2=−3/2,b1=0,b2=0,c=9,δ=4,andρ=−3/2, we obtain(41)F0=−12,F1=3ℨ22,F2=−ℨ4,F3=13ℨ630−ℨ42.Consequently, the semianalytical solution of equation (10) is written in the following form:(42)F=13ℨ630−3ℨ42+3ℨ22−12+⋯.


## 4.2. Cubic B-Spline Scheme

Employing the cubic B-spline scheme to evaluate the numerical solutions of equation (10). Using same initial and boundary conditions with respect to the obtained solutions (18) and (31), yields


## 5. Discussion

This section illustrates several of the results forFx,y,t to highlight the properties of the 2+1-dimensional integrable SKdV model with Miura transformation. In the follow-up, we fix the value of y to characterize these solutions and the interpretation is based on three different types of representations (three- and two-dimensional charts and contour plot). In the following steps, the physical interpretation of the represented figures is discussed:(i)
Figure1 shows the bright solitary for (18) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along x- axis, and the contour plot (c) to explain the overhead view of the solution when α=−1,a0=4,a−1=0,a−2=−3,c=72,δ=2,λ=0,μ=4,andρ=3(ii)
Figure2 shows the dark solitary for (31) in the three-dimensional plot (a) to illustrate the perspective view of the solution, the two-dimensional plot (b) to present the wave propagation pattern of the wave along the x-axis, and the contour plot (c) to explain the overhead view of the solution when a0=1,a1=0,a2=−3/2,b1=0,b2=0,c=9,δ=4,andρ=−3/2(iii)
Figure3 illustrates the exact and semianalytical obtained solutions, respectively, by the extended simplest equation method and Adomian decomposition method(iv)
Figure4 illustrates the exact and semianalytical obtained solutions, respectively, by the sech-tanh expansion method and Adomian decomposition method(v)
Figure5 illustrates the exact and numerical obtained solutions, respectively, via the sech-tanh expansion method and cubic B-spline scheme(vi)
Figure6 illustrates the exact and numerical obtained solutions, respectively, via the sech-tanh expansion method and cubic B-spline schemeNow, we shall show the accuracy of our obtained solution and explain the comparison between the two employed analytical schemes:(vii)
Tables1 and 2 show calculated values of the exact, semianalytical, and numerical solutions with different values of ℨ. These values show the accuracy of the obtained analytical solutions via the sech-tanh expansion method over the obtained analytical solutions via the extended simplest equation method where the absolute values of error in the sech-tanh method is smaller that those obtained by the extended simplest equation method. Figure 7 explains the absolute value of error in 1 and 2.Figure 1
Solitary wave in three different forms of equation (18).
(a)(b)(c)Figure 2
Solitary wave in three different forms of equation (31).
(a)(b)(c)Figure 3
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the extended simplest equation method.
(a)(b)(c)Figure 4
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the sech-tanh expansion method.
(a)(b)(c)Figure 5
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the extended simplest equation method.
(a)(b)(c)Figure 6
Exact and numerical solutions of equation (10) according to the obtained analytical solution via the sech-tanh expansion method.
(a)(b)(c)Figure 7
Absolute value of error via the Adomian decomposition method (a) and cubic B-spline scheme (b) based on the shown value in Tables1 and 2.
(a)(b)Table 1
Exact, semianalytical, and absolute values of error with different values ofℨ with respect to the obtained solution via extended simplest equation method (18) and sech-tanh method (31) via the Adomian decomposition method.
Ext. Sim Eq. methodSech-tanh methodAbsolute errorValue of 3ExactApproximateExactApproximateFirst schemeSecond scheme0.0011.999981.999990.4999990.4999991.19999E−054.99711×10−130.0021.99991.999950.4999940.4999940.0000479997.99988×10−120.0031.999781.999890.4999870.4999870.0001079954.05003×10−110.0041.999621.999810.4999760.4999760.0001919841.28001×10−100.0051.99941.99970.4999630.4999630.000299963.12502×10−100.0061.999141.999570.4999460.4999460.0004319176.48006×10−100.0071.998821.999410.4999270.4999270.0005878461.20052×10−90.0081.998461.999230.4999040.4999040.0007677382.04804×10−90.0091.998061.999030.4998790.4998790.000971583.28057×10−90.011.99761.99880.499850.499850.001199365.00013×10−9Table 2
Exact, numerical, and absolute value of error with different value ofℨ with respect to the obtained solution via the extended simplest equation method (18) and sech-tanh method (31) via the cubic B-spline scheme.
Value ofℨExt. Sim Eq. MethodSech-tanh methodAbsolute errorExactNumericalExactNumericalFirst schemeSecond scheme0−2−2−0.5−0.5000.001−1.99999−1.99987−0.499999−0.4999980.000125968.99958×10−120.002−1.99998−1.99971−0.499994−0.4999940.0002639191.59986×10−110.003−1.99995−1.99953−0.499987−0.4999870.0004138762.09974×10−110.004−1.9999−1.99933−0.499976−0.4999760.0005758292.39971×10−110.005−1.99985−1.9991−0.499963−0.4999630.0007497782.49963×10−110.006−1.99978−1.99885−0.499946−0.4999460.0009357212.39959×10−110.007−1.99971−1.99857−0.499927−0.4999270.001133662.09953×10−110.008−1.99962−1.99827−0.499904−0.4999040.001343581.59955×10−110.009−1.99951−1.99795−0.499879−0.4998790.00156558.99764×10−120.01−1.9994−1.9976−0.49985−0.499850.00179940


## 6. Conclusion

In this paper, the extended simplest equation and sech-tanh expansion methods have been successfully implemented on the2+1-dimensional integrable SKdV model with Miura transform. Moreover, the stability properties of the solutions have also been tackled. The Adomian decomposition method and cubic B-spline scheme have also employed to investigate the semianalytical and numerical solutions, and the two show the accuracy of the obtained analytical solutions. The rigor of the obtained solutions that have been obtained by the sech-tanh expansion method has been discussed. The solutions were represented by allowing a physical interpretation and better interpretation of their properties. In summary, this paper studied the SKdV and found relevant solutions that provide new interpretations of the real-world phenomena.



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