Multi-Symplectic Method for the Logarithmic-KdV Equation

Yu Zhang,Shaohua Li

doi:10.3390/sym12040545

Abstract

The multi-symplectic integrator, as a numerical integration approach with symmetry, is known to have the characteristic of preserving the qualitative features and geometric properties of certain systems. Using the multi-symplectic integrator, the numerical simulation of the Gaussian solitary wave propagation of the logarithmic Korteweg–de Vries (logarithmic-KdV) equation was investigated. The multi-symplectic formulation of the logarithmic-KdV equation was explored by introducing some intermediate variables. A fully implicit version of the centered box scheme was used to discretize the multi-symplectic equations. In addition, numerical experiments were carried out to show the conservative properties of the proposed scheme.

Highlights

Since being proposed by Bridges [1] and Marsden et al [2], multi-symplectic integrators have received huge attention in the past twenty years
In comparison with the well-known symplectic integrators used in Hamiltonian ordinary differential equations (ODEs), multi-symplectic integrators are able to overcome the limitation of symplectic geometry in dealing with partial differential equation (PDE) [1,3]
The symplecticness in symplectic structures is a global property, while the symplecticness in multi-symplectic structures is a local property which varies both in time and space [4]

Summary

Introduction

Since being proposed by Bridges [1] and Marsden et al [2], multi-symplectic integrators have received huge attention in the past twenty years. As a kind of geometric integration method, the multi-symplectic method aims to construct numerical schemes which can preserve qualitative features and geometric properties of the solution of a partial differential equation (PDE) under discretization. Gaussian solitary wave of the logarithmic-KdV equation numerically. The logarithmic-KdV equation, as a kind of generalized KdV equation that is derived as formal asymptotic limits of the nonlinear Fermi–Pasta–Ulam (FPU) lattices is defined by: ut + (u ln|u|)x + uxxx = 0 This equation describes the solitary wave propagation in harmonic chains with the nonlinear Hertzian interaction forces [20] that admit Gaussian solitary wave solutions. This work aims to extend the applications of the multi-symplectic integrator when dealing with the strong nonlinear traveling solitary wave. Some concluding remarks are given in the last section

A Multi-Symplectic Formulation for the Logarithmic-KdV Equation

A Multi-Symplectic Centered Box Scheme for the Logarithmic-KdV Equation

Numerical Experiments on Gaussian Solitary Wave Propagation

Conclusions