New rational solitary wave solutions of (2+1) -dimensional Burgers equation

Abstract

Based on the computer symbolic system Maple and the rational solitary wave solutions and rational periodic solutions obtained by others previously, we present a new direct ansätz in terms of the hyperbolic functions and periodic function to solve the nonlinear evolution equations, that is to say, we assume the equations have the solution in the following form u i ( ξ ) = a i 0 + ∑ j = 1 n i g j − 1 ( a i j f + b i j g + c i j h + d i j r ) , where f = tanh ( ξ ) sech ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) , g = sech ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) , h = sech 2 ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) , r = tanh ( ξ ) A + B tanh ( ξ ) + C sech 2 ( ξ ) and A 2 + B 2 + C 2 ≠ 0 . Compared with the solutions obtained by the existing tanh methods, Riccati equation method and all kinds of their improved methods, the solutions obtained by the proposed direct method not only include some obtained solutions, but also include some new and general rational solutions. And we choose the (2+1)-dimensional Burgers equation to illustrate the key step. As a result, we obtain abundant families of rational solitary wave solutions and rational periodic solutions. Furthermore, the direct method is proved to be efficient.