变形KdV方程孤立子及其相互作用的分析解与数值模拟

NUMERICAL MODELING OF SOLITONS OF MODIFIED KdV EQUATION AND THEIR INTERACTIONS BY A PETROV-GALERKIN FINITE ELEMENT METHOD

摘要: 本文讨论变形KdV方程的解法。首先借助Bäcklund变换的不变性定理和置换定理导出该方程孤立波传播及孤立子相互作用的分析解,然后用Petrov-Galerkin有限元法建立了求数值解的格式并以分析解提供的初值,数值地求解了该方程的Cauchy初值问题。所得数值解与分析解的高度吻合既印证了理论目的的正确性,也显示了数值目的的高精度和稳定性。

Abstract: This Paper deals with the initial value problems of the Modified KdV(MKdV) equation. First, the analytic solutions of solitary wave and interaction of two solitons are obtained by the Bäcklund Transformation. Then, a numerical scheme of a kind of Petrov-Galerkin finite element methods is derived and used to solve the initial value problems of MKdV equation numerically. Finally, the comparison of numerical and analytic solutions is made and shows the excellent accurancy and stability of the numerical scheme used in this paper.