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.

Automated summary

In this work, an exact solitary wave solution and a linear conservative difference scheme for the generalized KdV-K equation were proposed and rigorously proved for mass conservation, solvability, stability, and convergence. The scheme is second-order accurate in both time and space variables. The numerical method and theoretical analysis were extended to the 2D equation, with comparisons showing the efficiency and reliability of the proposed scheme.