Advances in Mathematical Physics
Volume 2019 (2019), Article ID 3175213, 5 pages
https://doi.org/10.1155/2019/3175213

Research Article

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

Abdullahi Rashid Adem,¹ Mohammad Mirzazadeh,² Qin Zhou,³ and Kamyar Hosseini⁴

¹Department of Mathematical Sciences, North-West University, Private Bag X 2046, Mmabatho 2735, South Africa
²Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar 44891-63157, Iran
³School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, China
⁴Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran

Correspondence should be addressed to Qin Zhou; qinzhou@whu.edu.cn

Received 16 January 2019; Accepted 14 March 2019; Published 2 May 2019

Academic Editor: Boris G. Konopelchenko

Copyright © 2019 Abdullahi Rashid Adem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

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 in which and are polynomials of exponential functions. The SK equation with a nonvanishing boundary condition [18, 19]

is 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]

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

to convert (1) and (2) into the following, respectively:

and

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:

Step 2. Suppose the solution of above NLEE can be expressed as

in which and are unknowns to be determined and

Step 3. Substituting (7) and its derivatives into (6) yields the following transformed equation:

Step 4. By setting the numerator of the function to zero, we will reach an algebraic system which its solution yields the multiple wave solution of (6) as

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

where is a constant and

the dispersion is

Now, applying the MEFS results in

one-soliton solution can be presented as

where is arbitrary but is defined by (13).

2.1.2. Two-Soliton Solution of (1)

To seek two-soliton solution, the following ansatz is considered

in which and are defined as

Now, by applying the MEFS, we acquire

2.1.3. Three-Soliton Solution of (1)

To derive three-soliton solution, it is assumed

in which and are defined as

Now, applying the MEFS yields

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

where is a constant and

and the dispersion relation is

Now, applying the MEFS results in

and, so, the resulting one-soliton solution can be presented as

where is arbitrary but is defined by (26).

2.2.2. Two-Soliton Solution of (2)

To seek two-soliton solution, the following ansatz is considered

in which and are defined as

Now, by applying the MEFS, we acquire

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

in which and are defined as

Now, applying the MEFS yields

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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work of Qin Zhou was supported by the National Natural Science Foundation of China (Grant Nos. 11705130 and 1157149), this author was also sponsored by the Chutian Scholar Program of Hubei Government in China.

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