Travelling wave solution of Dodd-Bullough-Mikhailov equation: a comparative study between Generalized Kudryashov and improved F-expansion methods

We investigate the efficiency of Generalized Kudryashov and improved F-expansion methods in solving nonlinear partial differential equations. The Dodd-Bullough-Mikhailov equation is considered to implement these methods. Both methods allow us to construct a number of travelling wave solutions of the governing equation. However, the Generalized Kudryashov method is found more direct, effective and requires less tedious symbolic computations compared to the improved F-expansion method. Our analysis also reveal that the basic version of either of the methods could be effective enough to acquire the fundamental wave solutions of the governing equation.


Introduction
The partial differential equation (PDE) is a useful tool for describing the phenomena that arise in mathematical physics and engineering. For example, heat flow, wave propagation, dispersion of chemically reactive material, fluid dynamics, quantum mechanics, electricity, plasma physics, propagation of shallow water waves, to name a few. Although linear partial differentials equations are sometimes used to model these problems, nonlinear equations are the best fit for these problems. Numerical solution techniques are often time used to solve these nonlinear equations. However, there has been ever-increasing interest from scientists and engineers in the analytical techniques for addressing these nonlinear problems as that is perhaps the most challenging, and promising area of modern mathematics.
However, because of a large number of methods, there is always a daunting problem of finding a suitable method for an equation under consideration. The purpose of this study is to find a suitable method for solving the Dodd-Bullough-Mikhailov equation (DBM) equation, which may serve as a guideline to look for a suitable method for other nonlinear equations. The DBM equation is a well-known nonlinear partial differential equation introduced by Roger Dodd, Robin Bullough and Alexander Mikhailov [40], which is a modified version of the Zhiber-Shabat equation obtained by setting = r 0. The DBM equation is closely connected with ( ) SL R 3, sigma-model describing the field triplet in the three-dimensional unimodular affine space [40,41]. Further, it appears in several other scientific applications, such as fluid dynamics, nonlinear optics, quantum field theory, electromagnetic waves and solid state physics. We solve the governing equation by using the Generalized Kudryashov and improved Fexpansion methods.
The remaining part of the paper is organized as follows. Section 2 introduces the working principle of the Generalized Kudryashov and improved F-expansion methods. Section 3 represents the solutions of the DBM equation. Section 4 summarizes the obtained results and compares the solution methods. Finally, section 5 concludes this work with the possible future research direction.

Methodology
A general nonlinear partial differential equation (NLEE) is written as is an unknown function and P is a polynomial of u and its various partial derivatives, in which the highest-order derivatives and nonlinear terms are involved. In order to construct the travelling wave solutions, we introduce the transformation , where x w =x t. This transformation converts equation (2.1) into an ordinary differential equation of the form Here w represents the speed of the travelling wave. The sign of w determines the direction of the wave, that is, whether the wave travels in a positive or negative direction. In the following, we briefly describe the main structures of the solution methods. We begin by introducing the well-known Riccati differential equation, which is often written as Depending on the value of the parameter k, the equation (2.3) has three different types of solutions, which are as follows: When < k 0, the governing equation possesses two equilibria, a stable node and unstable saddle point, which are born through saddle-node bifurcation. In this case, the solutions are When k changes from negative to zero, the equilibria that are born through saddle-node bifurcation get closer and finally collides when = k 0. The origin, in this case, is the only equilibrium which is semi-stable, and the solution has the following form Finally, when > k 0, the governing equation (2.3) has no equilibrium and hence, has the following two periodic solutions The integrating constant is assumed zero in all three cases above. The working steps of the Generalized Kudryashov and improved F-expansion methods are described in the following two subsections. [30][31][32] The main steps of Generalized Kudryashov method are as follows:

The generalized Kudryashov method
Step-1 Let us consider that the trial function of analytical solution for equation (2.2) as follows Step-2 We determine the positive integers N and M appearing in equation (2.1.1) by considering the homogeneous balance between the highest order derivatives and nonlinear terms in equation (2.2).
Step-3 Substituting equation (2.1.1) in equation (2.2) together with the value of N and M obtained in step 2, we get polynomials in We then set each coefficient of the resulting polynomial to zero to obtain a system of algebraic equations. The system of algebraic equations is then solved by using Maple to obtain the unknown parameters and w. We finally substitute the obtained value in equation (2.1.1) to construct the exact travelling wave solutions of equation (2.1).

The improved F-expansion method [33-36]
The main steps involved in F-expansion method are as follows: Step-1 Let us assume the traveling wave solution of equation (2.2) can be written as follows w and m are arbitrary constants to be determined, and Step-2 The positive integer N appearing in equation (2.2.1) can be obtained by taking the homogeneous balance between the highest order derivatives and nonlinear terms in equation (2.2). Step

Application of the methods
In this section, we apply the idea of the considered methods to solve the DBM equation. As mentioned in section 1, the DBM equation is given by , and x w =x t, we can recast the above equation as follows

The implementation of the generalized Kudryashov method
We apply the homogeneous balance between V 3 and  VV as described in Step-2 in section 2.1, which gives = + N M 2. Setting = M 1, we obtain = N 3. Hence, equation (2.1.1) becomes,  p q a a a a b b , , , , , , , , 0 1 2 3 0 1 and w, which gives the following set of solutions: ,  16  27  ,  1  3  ,  1  3  , , ,  Therefore, according to < k 0, > k 0, and = k 0, as described in section 2, we obtain the following families of solutions.
When < k 0, we get the following hyperbolic trigonometric solutions: When > k 0, we get the following trigonometric solutions: Finally, when = k 0, we get the following solution, Family 5 ln . According to < k 0, > k 0, and = k 0, as described in section 2, we obtain the following families of solutions.
When < k 0, we get the following hyperbolic trigonometric solutions:   When > k 0, we get the following trigonometric solutions:  ln  3  tan  4  tan  3 6 t a n ,   Finally, when = k 0, we get the following solutions: Family 9 Family 10 Family 11 ln . 19 2 2

Comparison between the methods
In this section, we summarize the obtained results and make a comparison between the two methods. Besides, we present the graphs of several solutions demonstrating well-known wave shapes. MATLAB surface plot with convergent mesh size for individual solution was used to produce these graphs.    suggest these three basic types of travelling wave solutions of the DBM equation, which justify the efficiency of both the methods that are considered in this study. Further, the above discussion reveals that the Generalized Kudryashov method produces less number of repeated solution. As a result, it was found more direct compared to the F-expansion method and required less tedious symbolic computations. However, if we could omit the repeated solutions produced by this method, the symbolic calculation is expected to be more straightforward. Similar to the basic Kudryashov method, a basic F-expansion method that can be obtained by setting = m 0 in equation (3.2.1), could also help us to omit the unnecessary solutions and reduce the volume of symbolic calculations. A further study showing a comparison between the Generalized Kudryashov and improved F-expansion methods with their simplified versions will be published elsewhere.

Conclusion
We investigated the travelling wave solutions of the DBM equation using Generalized Kudryashov and the improved F-expansion methods. Three basic types of wave solutions, such as bright singular solitons, dark singular solitons and periodic solutions were found. Several other repeated and singular solutions were also found. The existing studies, however, suggest that the basic solutions are the only fundamental wave solutions of governing equation. Therefore, the repeated and additional singular solutions are physically insignificant, thus ignored.
Among the two methods, the Generalized Kudryashov method was found more direct and produced less number of insignificant solutions, and the symbolic calculations were straightforward than the other. We, therefore, presume that if the methods are modified in such a way so that the extra solutions are omitted, the volume of symbolic calculations will be reduced significantly. The basic version of either of the methods could be a better alternative to get rid of these insignificant solutions. However, a further study showing a comparison between these methods with their simplified versions may help to justify this conclusion. Similar comparative studies on other nonlinear equations using the methods used in this study, or other methods if needed, may help to find the best effective method for the respective equation. and = k 1.