Solitary wave solution and a linear mass-conservative difference scheme for the generalized Korteweg–de Vries–Kawahara equation

In this work, an exact solitary wave solution and a linear conservative difference scheme for solving the generalized Korteweg–de Vries–Kawahara (GKdV-K) equation are proposed. We first use the Ansatz’s method to derive the exact solitary wave solution for the GKdV-K equation and then develop a three-level linear conservative finite difference scheme for solving the equation. The mass conservation, solvability, stability and convergence of the numerical solution are rigorously proved. The scheme is second-order accurate in both time and space variables. We further extend the numerical method and theoretical analysis to the 2D GKdV-K equation. Comparisons between the solutions obtained from the exact solitary wave solution and the linear finite difference scheme are made to demonstrate that the present scheme is efficient and reliable.