A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

Abstract

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

Application of the Generalized Simplest Equation Method to the Burgers Equation

“…We have given twelve kinds of exact solutions for Equation (11) in literature [20]. We substitute these solutions ( )…”

“…By the same manner, from the rest two solutions of Equation (29) which are given in literature [20], we should obtain additional exact solutions in different styles to the Burgers equation. We have also obtained the multi-soliton solutions and the rational solutions of the Burgers equation by using the method.…”

“…For obtaining more general form exact solutions of NLEEs, recently we introduced a new direct method called the generalized simplest equation method to look for exact solutions of NLEEs [20]. In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation.…”

“…These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions and other types of solutions. This method has its own advantages [20] which are different from other methods, and it can derive new exact solutions of NLEEs which are not obtained by using other methods [5]- [14]. In this paper, we will use this method to construct more exact solutions of the Burgers equation.…”