Traveling wave solutions of the generalized nonlinear evolution equations
Introduction
In this paper we present a method, which enables one to construct traveling wave solutions for the following family of nonlinear partial differential equations:The equation is of order N + 1 and depends on N + 1 parameters denoted by α1, …, αN, m. This family contains several famous partial differential equations, such as the Korteweg–de Vries equation [1], [2], [3], [4], [5], the Korteweg–de Vries–Burgers equation [6], the Kuramoto–Sivashinsky equation [7], [8], [9], [10], [11], [12] and their generalizations [13], [14], [15], [16], [17]. Various members of the family (1.1) at N = 3 have been intensively studied in recent years. For example, see [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].
Our approach is based on constructing Bäcklund transformations between the traveling wave reductions of the equation and the Nth-order ordinary differential equation with two remarkable properties. First of all, this equation no longer contains the terms with m in the exponents, i.e. the parameter m moves from the exponents to the coefficients. Second, its mero-morphic solutions have only simple poles. All these makes the equation in question extremely convenient for further analysis. In particular, this equation admits application of various methods for constructing exact solutions. In this paper, we will use the simplest equation method suggested and developed in works [29], [30], [31], [32], [33] accordingly. The simplest equation method is based on expressing exact solutions of the given equation through solutions of another equation, called the simplest equation.
This paper is organized as follows. In Section 2, we describe our method: we derive the Bäckhand transformations and introduce the simplest equation. In Sections 3 Solitary wave solutions of the generalized Burgers equation, 4 Solitary wave solutions of the generalized Korteweg–de Vries–Burgers equation, 5 Solitary wave solutions of the generalized Kuramoto–Sivashinsky equation, 6 Solitary wave solutions of the fifth-order equation, we give several specific examples for the most interesting equations of the family (1.1).
Section snippets
Method applied
First of all let us show that there exist Bäckhand transformations between the traveling wave reductions of the equation and another ordinary differential equation with the properties discussed in Section 1. Making the substitution:in the equation and integrating the result yields the following Nth-order equation:provided that m ≠ −1. The constant of integration is set to be zero. Suppose y(z) is a
Solitary wave solutions of the generalized Burgers equation
The first member of family (1.1) is the generalized Burgers equation:This equation possesses the traveling wave reduction (2.1) with y(z) satisfying the equationEq. (3.2) is the Bernulli equation. The equalitiesprovide Bäckhand transformations between Bernulli equation (3.2) and the Riccati equation:Following the procedure suggested in the previous section we find the coefficients a and b:The general
Solitary wave solutions of the generalized Korteweg–de Vries–Burgers equation
Let us look for exact solutions of the generalized Korteweg–de Vries–Burgers equation:Using traveling wave reduction (2.1), we obtain the following second-order ordinary differential equation:The Bäcklund transformations:relate solutions of Eq. (4.2) and of the equation:The latter equation possesses solutions of the type (2.10) withprovided that:
Solitary wave solutions of the generalized Kuramoto–Sivashinsky equation
In this section, we will construct exact solutions of the generalized Kuramoto–Sivashinsky equation:This equation possesses the traveling wave reduction (2.1) with y(z) satisfying the equation:The Bäcklund transformations:relate solutions of this equation and of the equation:Following the procedure
Solitary wave solutions of the fifth-order equation
In this section, we consider the following nonlinear fifth-order evolution equation:Again making the substitution (2.1) in (6.1) and integrating the result one getsEq. (2.7) for the function w(z) can be written asThis equation in the case m ≠ −8/3
Conclusion
In this paper, we have introduced an algorithm of finding traveling wave solutions for a family of nonlinear partial differential equation (1.1). This family contains a number of equations with mathematical and physical applications. The algorithm is based on constructing Bäcklund transformations between the traveling wave reductions of (1.1) and ordinary differential equations more simple for further analysis.
We would like to note that the same procedure can be applied to the equation
Acknowledgement
This work was supported by the International Science and Technology Center under Project B 1213.
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