Nonlinear dispersive waves in the discrete modified KdV equation

TL;DR

This paper investigates nonlinear dispersive waves in the discrete modified KdV equation using numerical simulations, quasi-continuum models, Whitham analysis, and compares theoretical predictions with numerical results.

Contribution

It introduces new quasi-continuum models and analytical methods to approximate and analyze dispersive wave structures in the discrete mKdV equation.

Findings

Quasi-continuum models accurately approximate dispersive wave profiles.

Whitham analysis provides insights into edge characteristics of dispersive shock waves.

Numerical results confirm the effectiveness of the proposed models.

Abstract

In this paper, we study the nonlinear dispersive waves including the rarefaction and dispersive shock waves in the discrete modified KdV equation through the numerical simulations of the dispersive Riemann problems. In particular, we propose distinct quasi-continuum models to approximate both the spatial profiles and distinct edge features of these two specific dispersive wave structures. Whitham analysis is performed to construct a closed system of partial differential equations which describe the slowly-varying dynamics of all the relevant parameters associated with the periodic traveling waves of the proposed quasi-continuum models. We then perform reduction on such modulation system to obtain a system of two simple-wave ordinary differential equations which lead to the DSW-fitting method that shall provide useful theoretical insights on different edge characteristics of the dispersive shock waves. Furthermore, we compute analytically the self-similar solutions corresponding to the dispersionless systems of the quasi-continuum models, which can be utilized to approximate the numerically observed rarefaction waves of the discrete mKdV equation. A systematic numerical comparison of these theoretical findings with their associated numerical counterparts finally demonstrate the good performance of the proposed quasi-continuum models in approximating both nonlinear dispersive wave patterns.