New analytical solutions by the application of the modified double sub-equation method to the (1 + 1)-Schamel-KdV equation, the Gardner equation and the Burgers equation

Abstract

In this research we present the application of the modified double sub-equation guess solution together with the analytical solutions of the Riccati equation to obtain new analytical exact solutions to the (1 + 1)-Schamel-KdV equation, the (1 + 1)-dimensional Gardner equation (or combined KdV-mKdV) and the nonlinear evolution (1 + 1)-dimensional Burgers equation. Results show some conditions between the allowed values of the interaction coefficients and the parameters of the allowed analytical solutions of the double sub-equation guess solution. The resulting new soliton solutions have by first time the proper real phase value behavior, not seen in previous applications of the double sub-equation method. Additionally, it has been revealed that handle the complexiton process with two different traveling variables have as an important consequence that the mixing between trigonometric and hyperbolic solutions is only observed for the analytical solutions of the nonlinear evolution (1 + 1)-dimensional Burgers equation, while no mixing between trigonometric and hyperbolic solutions is observed for the solutions of the (1 + 1)-Schamel-KdV ((1 + 1)-S-KdV) equation or the (1 + 1)-dimensional Gardner equation. Numerical simulations are provided in 3-D graphs and 2-D plots representing the general characteristics of the resulting analytical solutions.