Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives
Introduction
Traveling waves exist in many natural systems. Because of this traveling wave solutions of the nonlinear partial differential equations are studied much in the last decades [1], [2], [3], [4], [5] and effective methods for obtaining such solutions for integrable systems are developed [6], [7]. Our discussion below will be based on a specific approach for obtaining exact special solutions of nonlinear PDEs: the method of simplest equation and its version called modified method of simplest equation [8], [9], [10], [11]. The method of simplest equation is based on a procedure analogous to the first step of the test for the Painleve property [12], [13], [14]. In the version of the method called modified method of the simplest equation [10], [11] this procedure is substituted by the concept for the balance equation. This version of the method of simplest equation has been successfully applied for obtaining exact traveling wave solutions of numerous nonlinear PDEs such as versions of generalized Kuramoto–Sivashinsky equation, reaction–diffusion equation, reaction-telegraph equation [10], [15] generalized Swift–Hohenberg equation and generalized Rayleigh equation [11], extended Korteweg–de Vries equation [16], etc. [17], [18].
The method of simplest equation works as follows. By means of an appropriate ansatz the solved nonlinear PDE is reduced to a nonlinear ODE
Then the finite-series solution is substituted in (1). are coefficients and is solution of simpler ordinary differential equation called simplest equation. Let the result of this substitution be a polynomial of . is a solution of Eq. (1) if all coefficients of the obtained polynomial of are equal to 0. This condition leads to a system of nonlinear algebraic equations. Each non-trivial solution of this system corresponds to a solution of the studied nonlinear PDE.
In the last years we observe a fast development of tools that are connected to the method of simplest equation. These tools are effectively used for obtaining exact solutions of many nonlinear differential equations. As an example we mention [19] where in a systematic way on the basis of the Q-expansion method of Kudryashov and by using the concepts of ordinary and differential monomials and Newton polygons the authors obtained the forms of many nonlinear differential equations that have given kind of solution. Similar idea will be used below.
We shall consider traveling-wave solutions constructed on the basis of the simplest equationwhich solution is . and are parameters. Let us note here that the transformation transforms Eq. (2) to the equation . The last equation is a version of the Riccati equation which is much used as a simplest equation in the applications of the method of simplest equation [20], [21].
In addition we shall use the following concept of grade of monomial with respect to participating derivatives. Let us consider polynomials that are linear combination of monomials where each monomial contains product of terms consisting of powers of derivatives of different orders. This product of terms can be multiplied by a polynomial of u. Let a term from a monomial contain kth power of a derivative of lth order. We shall call the product kl grade of the term with respect to participating derivatives. The sum of these grades of all terms of a monomial will be called grade of the monomial with respect to participating derivatives. The general case is (1): The polynomial contains monomials that contain derivatives have odd or even grades with respect to participating derivatives. There are two particular cases: (1A): All monomials that contain derivatives have odd grades with respect to participating derivatives; (1B): All monomials that contain derivatives have even grades with respect to participating derivatives. We shall formulate our main result for the general case (1) and then we shall demonstrate several applications for the cases (1), (1A), (1B).
Section snippets
Main result
Below we search for solitary wave solutions for the class of nonlinear PDEs that contain monomials of derivatives which order with respect to participating derivatives is even and monomials of derivatives which order with respect to participating derivatives is odd. Theorem Let be a polynomial of the function and its derivatives. belongs to the differentiability class , where k is the highest order of derivative participating in . can contain some or all of the following parts: (A)
The two particular classes (1A) and (1B)
For the class of nonlinear PDEs that contain monomials of derivatives which order with respect to participating derivatives is odd only (particular class (1A)) the above theorem is valid when there are no terms of kind and . Famous example of nonlinear PDE of this class is the Korteweg–de Vries equationwhere K is a constant (in the original equation ). Eq. (12) contains three monomials and all of them have odd grade with respect to the participating derivatives. The
Concluding remarks
In this article we formulate an approach for obtaining solitary wave solutions of a large class of nonlinear PDEs. The discussed solitary wave solutions have 3 parameters: , and . If the system of nonlinear algebraic equations contains 3 equations then these 3 parameters can be expressed by the parameters of the nonlinear PDE and there will be no need of relationships between the parameters of the nonlinear PDE. If the system of nonlinear algebraic equations contain more than 3 equations
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