Exact single traveling wave solutions to generalized (2+1)-dimensional Gardner equation with variable coefficients

Abstract

We firstly apply the trial equation method to generalized (2+1)-dimensional Gardner equation to reduce the nonlinear partial differential equation into ordinary equation. Then by the complete discrimination system for polynomial method, the classification of the exact single traveling wave solutions is presented. This is the first time that the two methods are applied to integral-differential equation. In the classification, we can see that the equation has rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions solutions with double periodic, which are impossible to be obtained by other methods. Moreover, to ensure the existences of these solutions, concrete examples are also constructed with specific parameters. To the best of our knowledge, the results presented in the paper are brand new and can not be found in any other papers.