# Multiple Soliton Solutions of the Sawada-Kotera Equation with a Nonvanishing Boundary Condition and the Perturbed Korteweg de Vries Equation by Using the Multiple Exp-Function Scheme

Authors: Abdullahi Rashid Adem; Mohammad Mirzazadeh; Qin Zhou; Kamyar Hosseini
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

The Sawada-Kotera equation with a nonvanishing boundary condition, which models the evolution of steeper waves of shorter wavelength than those depicted by the Korteweg de Vries equation, is analyzed and also the perturbed Korteweg de Vries (pKdV) equation. For this goal, a capable method known as the multiple exp-function scheme (MEFS) is formally utilized to derive the multiple soliton solutions of the models. The MEFS as a generalization of Hirota’s perturbation method actually suggests a systematic technique to handle nonlinear evolution equations (NLEEs).


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## Body

## 1. Introduction

In the applied science, NLEEs are extensively used in theoretical studies to model a wide range of nonlinear phenomena. To comprehend the mechanisms of nonlinear phenomena, it is vital to investigate the solutions of NLEEs [1–15]. One specific tool that has recently achieved a special interest from academic researchers is the multiple exp-function scheme [16, 17]. The MEFS supposes that the multisoliton solutions of NLEEs can be presented as u(x,t)=p/q in which p and q are polynomials of exponential functions. The SK equation with a nonvanishing boundary condition [18, 19](1)ut+aa5bu+3u2+uxxx+b15u3+15uuxx+uxxxxx=0is one of NLEEs that models the evolution of steeper waves of shorter wavelength than those explained by the KdV equation and its perturbed form. By using the binary-Bell-polynomial Hirota method and symbolic computation, the bilinear form and N-soliton solutions for this model were derived in [19]. The perturbed form of KdV equation [18, 20–22](2)ut+6uux+uxxx+ϵ30u2ux+10uuxxx+20uxuxx+uxxxxx=0,is another kind of NLEEs describing some arrays of wave crests. The bilinear form, Bäcklund transformation, superposition formulae, and N-soliton solutions in terms of the Wronskian were done in [22].For computational purposes, we use the transformation(3)u=vxto convert (1) and (2) into the following, respectively:(4)vxt+aa5bvxx+6vxvxx+vxxxx+b45vx2vxx+15vxvxxxx+15vxxvxxx+vxxxxxx=0and(5)vxt+6vxvxx+vxxxx+ϵ30vx2vxx+10vxvxxxx+20vxxvxxx+vxxxxxx=0.The key goal of present work is applying the MEFS to generate the multiple soliton solutions of the models (1) and (2).


## 2. Multiple Exp-Function Method

The key steps of MEFS can be summarized as follows [16, 17].Step 1.
Let us consider the following (1 + 1)-dimensional NLEE:(6)Px,t,ux,ut,⋯=0.Step 2.
Suppose the solution of above NLEE can be expressed as(7)ux,t=pη1,η2,⋯,ηnqη1,η2,⋯,ηn,p=∑r,s=1n∑i,j=0Mprs,ijηriηsj,q=∑r,s=1n∑i,j=0Nqrs,ijηriηsj,
in whichprs,ij and qrs,ij are unknowns to be determined and(8)ηi=cieξi,ξi=kix-ωit,1⩽i⩽n,Step 3.
Substituting (7) and its derivatives into (6) yields the following transformed equation:(9)Qx,t,η1,η2,⋯,ηn=0.Step 4.
By setting the numerator of the functionQ(x,t,η1,η2,⋯,ηn) to zero, we will reach an algebraic system which its solution yields the multiple wave solution of (6) as(10)ux,t=pc1ek1x-ω1t,⋯,cneknx-ωntqc1ek1x-ω1t,⋯,cneknx-ωnt.


### 2.1. Multiple Soliton Solutions of SK Equation with a Nonvanishing Boundary Condition

In the current subsection, the multiple soliton solutions of SK equation with a nonvanishing boundary condition are derived through the MEFS.


#### 2.1.1. One-Soliton Solution of (1)

To obtain one-soliton solution, it is assumed(11)vx,t=pq,p=A1eθ1,q=1+eθ1,where A1 is a constant and(12)θ1=k1x-ω1tthe dispersion is(13)ω1=k15b2k14+5abk12+a25b.Now, applying the MEFS results in(14)A1=2k1,one-soliton solution can be presented as(15)vx,t=2k1ek1x-ω1t1+ek1x-ω1twhere k1 is arbitrary but ω1 is defined by (13).


#### 2.1.2. Two-Soliton Solution of (1)

To seek two-soliton solution, the following ansatz is considered(16)vx,t=pqin which p and q are defined as(17)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2A12k1+k2ek1x-ω1tek2x-ω2t,(18)q=1+ek1x-ω1t+ek2x-ω2t+A12ek1x-ω1tek2x-ω2t.Now, by applying the MEFS, we acquire(19)A12=5bk14-15bk13k2+20bk12k22-15bk1k23+5bk24+3ak12-6ak1k2+3ak225bk14+15bk13k2+20bk12k22+15bk1k23+5bk24+3ak12+6ak1k2+3ak22,ω1=k15b2k14+5abk12+a25b,ω2=k25b2k24+5abk22+a25b.


#### 2.1.3. Three-Soliton Solution of (1)

To derive three-soliton solution, it is assumed(20)vx,t=pqin which p and q are defined as(21)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2k3ek3x-ω3t+2A12k1+k2e-ω1t+k1xe-ω2t+k2x+2A13k1+k3ek1x-ω1tek3x-ω3t+2A23k2+k3ek2x-ω2tek3x-ω3t+2A12A13A23k1+k2+k3ek1x-ω1tek2x-ω2tek3x-ω3t,(22)q=1+ek1x-ω1t+ek2x-ω2t+ek3x-ω3t+A12ek1x-ω1te-ω2t+k2x+A13ek1x-ω1tek3x-ω3t+A23ek2x-ω2tek3x-ω3t+A12A13A23ek1x-ω1tek2x-ω2tek3x-ω3t.Now, applying the MEFS yields(23)A12=5bk14-15bk13k2+20bk12k22-15bk1k23+5bk24+3ak12-6ak1k2+3ak225bk14+15bk13k2+20bk12k22+15bk1k23+5bk24+3ak12+6ak1k2+3ak22,A13=5bk14-15bk13k3+20bk12k32-15bk1k33+5bk34+3ak12-6ak1k3+3ak325bk14+15bk13k3+20bk12k32+15bk1k33+5bk34+3ak12+6ak1k3+3ak32,A23=5bk24-15bk23k3+20bk22k32-15bk2k33+5bk34+3ak22-6ak2k3+3ak325bk24+15bk23k3+20bk22k32+15bk2k33+5bk34+3ak22+6ak2k3+3ak32,ω1=k15b2k14+5abk12+a25b,ω2=k25b2k24+5abk22+a25b,ω3=k35b2k34+5abk32+a25b.


### 2.2. Multiple Soliton Solutions of pKdV Equation (2)

In the present subsection, the multiple soliton solutions of pKdV equation are obtained through the MEFS.


#### 2.2.1. One-Soliton Solution of (2)

To obtain one-soliton solution, it is assumed(24)vx,t=pq,p=A1eθ1,q=1+eθ1,where A1 is a constant and(25)θ1=k1x-ω1tand the dispersion relation is(26)ω1=ϵk15+k13.Now, applying the MEFS results in(27)A1=2k1and, so, the resulting one-soliton solution can be presented as(28)vx,t=2k1ek1x-ω1t1+ek1x-ω1t,where k1 is arbitrary but ω1 is defined by (26).


#### 2.2.2. Two-Soliton Solution of (2)

To seek two-soliton solution, the following ansatz is considered(29)vx,t=pqin which p and q are defined as(30)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2A12k1+k2ek1x-ω1tek2x-ω2t,(31)q=1+ek1x-ω1t+ek2x-ω2t+A12ek1x-ω1tek2x-ω2t.Now, by applying the MEFS, we acquire(32)A12=k12-2k1k2+k22k12+2k1k2+k22,ω1=ϵk15+k13,ω2=ϵk25+k23.A profile of the evolution of the two-soliton solution (29) is given in Figure 1.Figure 1
Evolution of the two-soliton solution (29).


#### 2.2.3. Three-Soliton Solution of (2)

To derive three-soliton solution, it is assumed(33)vx,t=pqin which p and q are defined as(34)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2k3ek3x-ω3t+2A12k1+k2e-ω1t+k1xe-ω2t+k2x+2A13k1+k3ek1x-ω1tek3x-ω3t+2A23k2+k3ek2x-ω2tek3x-ω3t+2A12A13A23k1+k2+k3ek1x-ω1tek2x-ω2tek3x-ω3t,(35)q=1+ek1x-ω1t+ek2x-ω2t+ek3x-ω3t+A12ek1x-ω1te-ω2t+k2x+A13ek1x-ω1tek3x-ω3t+A23ek2x-ω2tek3x-ω3t+A12A13A23ek1x-ω1tek2x-ω2tek3x-ω3t.Now, applying the MEFS yields(36)A12=k12-2k1k2+k22k12+2k1k2+k22,A13=k12-2k1k3+k32k12+2k1k3+k32,A23=k22-2k2k3+k32k22+2k2k3+k32,ω1=ϵk15+k13,ω2=ϵk25+k23,ω3=ϵk35+k33.A profile of the evolution of the three-soliton solution (33) is given in Figure 2.Figure 2
Evolution of the three-soliton solution (33).


## 2.1. Multiple Soliton Solutions of SK Equation with a Nonvanishing Boundary Condition

In the current subsection, the multiple soliton solutions of SK equation with a nonvanishing boundary condition are derived through the MEFS.


### 2.1.1. One-Soliton Solution of (1)

To obtain one-soliton solution, it is assumed(11)vx,t=pq,p=A1eθ1,q=1+eθ1,where A1 is a constant and(12)θ1=k1x-ω1tthe dispersion is(13)ω1=k15b2k14+5abk12+a25b.Now, applying the MEFS results in(14)A1=2k1,one-soliton solution can be presented as(15)vx,t=2k1ek1x-ω1t1+ek1x-ω1twhere k1 is arbitrary but ω1 is defined by (13).


### 2.1.2. Two-Soliton Solution of (1)

To seek two-soliton solution, the following ansatz is considered(16)vx,t=pqin which p and q are defined as(17)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2A12k1+k2ek1x-ω1tek2x-ω2t,(18)q=1+ek1x-ω1t+ek2x-ω2t+A12ek1x-ω1tek2x-ω2t.Now, by applying the MEFS, we acquire(19)A12=5bk14-15bk13k2+20bk12k22-15bk1k23+5bk24+3ak12-6ak1k2+3ak225bk14+15bk13k2+20bk12k22+15bk1k23+5bk24+3ak12+6ak1k2+3ak22,ω1=k15b2k14+5abk12+a25b,ω2=k25b2k24+5abk22+a25b.


### 2.1.3. Three-Soliton Solution of (1)

To derive three-soliton solution, it is assumed(20)vx,t=pqin which p and q are defined as(21)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2k3ek3x-ω3t+2A12k1+k2e-ω1t+k1xe-ω2t+k2x+2A13k1+k3ek1x-ω1tek3x-ω3t+2A23k2+k3ek2x-ω2tek3x-ω3t+2A12A13A23k1+k2+k3ek1x-ω1tek2x-ω2tek3x-ω3t,(22)q=1+ek1x-ω1t+ek2x-ω2t+ek3x-ω3t+A12ek1x-ω1te-ω2t+k2x+A13ek1x-ω1tek3x-ω3t+A23ek2x-ω2tek3x-ω3t+A12A13A23ek1x-ω1tek2x-ω2tek3x-ω3t.Now, applying the MEFS yields(23)A12=5bk14-15bk13k2+20bk12k22-15bk1k23+5bk24+3ak12-6ak1k2+3ak225bk14+15bk13k2+20bk12k22+15bk1k23+5bk24+3ak12+6ak1k2+3ak22,A13=5bk14-15bk13k3+20bk12k32-15bk1k33+5bk34+3ak12-6ak1k3+3ak325bk14+15bk13k3+20bk12k32+15bk1k33+5bk34+3ak12+6ak1k3+3ak32,A23=5bk24-15bk23k3+20bk22k32-15bk2k33+5bk34+3ak22-6ak2k3+3ak325bk24+15bk23k3+20bk22k32+15bk2k33+5bk34+3ak22+6ak2k3+3ak32,ω1=k15b2k14+5abk12+a25b,ω2=k25b2k24+5abk22+a25b,ω3=k35b2k34+5abk32+a25b.


## 2.1.1. One-Soliton Solution of (1)

To obtain one-soliton solution, it is assumed(11)vx,t=pq,p=A1eθ1,q=1+eθ1,where A1 is a constant and(12)θ1=k1x-ω1tthe dispersion is(13)ω1=k15b2k14+5abk12+a25b.Now, applying the MEFS results in(14)A1=2k1,one-soliton solution can be presented as(15)vx,t=2k1ek1x-ω1t1+ek1x-ω1twhere k1 is arbitrary but ω1 is defined by (13).


## 2.1.2. Two-Soliton Solution of (1)

To seek two-soliton solution, the following ansatz is considered(16)vx,t=pqin which p and q are defined as(17)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2A12k1+k2ek1x-ω1tek2x-ω2t,(18)q=1+ek1x-ω1t+ek2x-ω2t+A12ek1x-ω1tek2x-ω2t.Now, by applying the MEFS, we acquire(19)A12=5bk14-15bk13k2+20bk12k22-15bk1k23+5bk24+3ak12-6ak1k2+3ak225bk14+15bk13k2+20bk12k22+15bk1k23+5bk24+3ak12+6ak1k2+3ak22,ω1=k15b2k14+5abk12+a25b,ω2=k25b2k24+5abk22+a25b.


## 2.1.3. Three-Soliton Solution of (1)

To derive three-soliton solution, it is assumed(20)vx,t=pqin which p and q are defined as(21)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2k3ek3x-ω3t+2A12k1+k2e-ω1t+k1xe-ω2t+k2x+2A13k1+k3ek1x-ω1tek3x-ω3t+2A23k2+k3ek2x-ω2tek3x-ω3t+2A12A13A23k1+k2+k3ek1x-ω1tek2x-ω2tek3x-ω3t,(22)q=1+ek1x-ω1t+ek2x-ω2t+ek3x-ω3t+A12ek1x-ω1te-ω2t+k2x+A13ek1x-ω1tek3x-ω3t+A23ek2x-ω2tek3x-ω3t+A12A13A23ek1x-ω1tek2x-ω2tek3x-ω3t.Now, applying the MEFS yields(23)A12=5bk14-15bk13k2+20bk12k22-15bk1k23+5bk24+3ak12-6ak1k2+3ak225bk14+15bk13k2+20bk12k22+15bk1k23+5bk24+3ak12+6ak1k2+3ak22,A13=5bk14-15bk13k3+20bk12k32-15bk1k33+5bk34+3ak12-6ak1k3+3ak325bk14+15bk13k3+20bk12k32+15bk1k33+5bk34+3ak12+6ak1k3+3ak32,A23=5bk24-15bk23k3+20bk22k32-15bk2k33+5bk34+3ak22-6ak2k3+3ak325bk24+15bk23k3+20bk22k32+15bk2k33+5bk34+3ak22+6ak2k3+3ak32,ω1=k15b2k14+5abk12+a25b,ω2=k25b2k24+5abk22+a25b,ω3=k35b2k34+5abk32+a25b.


## 2.2. Multiple Soliton Solutions of pKdV Equation (2)

In the present subsection, the multiple soliton solutions of pKdV equation are obtained through the MEFS.


### 2.2.1. One-Soliton Solution of (2)

To obtain one-soliton solution, it is assumed(24)vx,t=pq,p=A1eθ1,q=1+eθ1,where A1 is a constant and(25)θ1=k1x-ω1tand the dispersion relation is(26)ω1=ϵk15+k13.Now, applying the MEFS results in(27)A1=2k1and, so, the resulting one-soliton solution can be presented as(28)vx,t=2k1ek1x-ω1t1+ek1x-ω1t,where k1 is arbitrary but ω1 is defined by (26).


### 2.2.2. Two-Soliton Solution of (2)

To seek two-soliton solution, the following ansatz is considered(29)vx,t=pqin which p and q are defined as(30)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2A12k1+k2ek1x-ω1tek2x-ω2t,(31)q=1+ek1x-ω1t+ek2x-ω2t+A12ek1x-ω1tek2x-ω2t.Now, by applying the MEFS, we acquire(32)A12=k12-2k1k2+k22k12+2k1k2+k22,ω1=ϵk15+k13,ω2=ϵk25+k23.A profile of the evolution of the two-soliton solution (29) is given in Figure 1.Figure 1
Evolution of the two-soliton solution (29).


### 2.2.3. Three-Soliton Solution of (2)

To derive three-soliton solution, it is assumed(33)vx,t=pqin which p and q are defined as(34)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2k3ek3x-ω3t+2A12k1+k2e-ω1t+k1xe-ω2t+k2x+2A13k1+k3ek1x-ω1tek3x-ω3t+2A23k2+k3ek2x-ω2tek3x-ω3t+2A12A13A23k1+k2+k3ek1x-ω1tek2x-ω2tek3x-ω3t,(35)q=1+ek1x-ω1t+ek2x-ω2t+ek3x-ω3t+A12ek1x-ω1te-ω2t+k2x+A13ek1x-ω1tek3x-ω3t+A23ek2x-ω2tek3x-ω3t+A12A13A23ek1x-ω1tek2x-ω2tek3x-ω3t.Now, applying the MEFS yields(36)A12=k12-2k1k2+k22k12+2k1k2+k22,A13=k12-2k1k3+k32k12+2k1k3+k32,A23=k22-2k2k3+k32k22+2k2k3+k32,ω1=ϵk15+k13,ω2=ϵk25+k23,ω3=ϵk35+k33.A profile of the evolution of the three-soliton solution (33) is given in Figure 2.Figure 2
Evolution of the three-soliton solution (33).


## 2.2.1. One-Soliton Solution of (2)

To obtain one-soliton solution, it is assumed(24)vx,t=pq,p=A1eθ1,q=1+eθ1,where A1 is a constant and(25)θ1=k1x-ω1tand the dispersion relation is(26)ω1=ϵk15+k13.Now, applying the MEFS results in(27)A1=2k1and, so, the resulting one-soliton solution can be presented as(28)vx,t=2k1ek1x-ω1t1+ek1x-ω1t,where k1 is arbitrary but ω1 is defined by (26).


## 2.2.2. Two-Soliton Solution of (2)

To seek two-soliton solution, the following ansatz is considered(29)vx,t=pqin which p and q are defined as(30)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2A12k1+k2ek1x-ω1tek2x-ω2t,(31)q=1+ek1x-ω1t+ek2x-ω2t+A12ek1x-ω1tek2x-ω2t.Now, by applying the MEFS, we acquire(32)A12=k12-2k1k2+k22k12+2k1k2+k22,ω1=ϵk15+k13,ω2=ϵk25+k23.A profile of the evolution of the two-soliton solution (29) is given in Figure 1.Figure 1
Evolution of the two-soliton solution (29).


## 2.2.3. Three-Soliton Solution of (2)

To derive three-soliton solution, it is assumed(33)vx,t=pqin which p and q are defined as(34)p=2k1ek1x-ω1t+2k2ek2x-ω2t+2k3ek3x-ω3t+2A12k1+k2e-ω1t+k1xe-ω2t+k2x+2A13k1+k3ek1x-ω1tek3x-ω3t+2A23k2+k3ek2x-ω2tek3x-ω3t+2A12A13A23k1+k2+k3ek1x-ω1tek2x-ω2tek3x-ω3t,(35)q=1+ek1x-ω1t+ek2x-ω2t+ek3x-ω3t+A12ek1x-ω1te-ω2t+k2x+A13ek1x-ω1tek3x-ω3t+A23ek2x-ω2tek3x-ω3t+A12A13A23ek1x-ω1tek2x-ω2tek3x-ω3t.Now, applying the MEFS yields(36)A12=k12-2k1k2+k22k12+2k1k2+k22,A13=k12-2k1k3+k32k12+2k1k3+k32,A23=k22-2k2k3+k32k22+2k2k3+k32,ω1=ϵk15+k13,ω2=ϵk25+k23,ω3=ϵk35+k33.A profile of the evolution of the three-soliton solution (33) is given in Figure 2.Figure 2
Evolution of the three-soliton solution (33).


## 3. Concluding Remarks

We note that multiple soliton solutions of (1) and (2) are in agreement with [19, 21, 22]. Moreover, we should emphasize the approach employed here was independent of the bilinear forms, simplified Hirota method, Darboux transformation method, or Bell polynomial technique. Finally we can say the multiple exp-function algorithm is an elegant and versatile method that can be adopted to other NLEEs of mathematical physics.



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