Nonlinear dispersive waves in the discrete modified KdV equation

Abstract In this paper, we study the nonlinear dispersive waves including the numerically observed rarefaction and dispersive shock waves in the discrete modified KdV equation which has widespread applications in plasma physics. In particular, we propose three distinct quasi-continuum models to approximate both the spatial profiles and distinct edge features of these specific dispersive wave structures. Whitham analysis is performed to construct a closed system of partial differential equations which describes the slowly-varying dynamics of all the relevant parameters associated with the periodic travelling waves of the proposed quasi-continuum models. We then apply the DSW-fitting method that shall provide useful theoretical insights into different edge characteristics of the dispersive shock waves. Furthermore, we compute analytically the self-similar solutions corresponding to the dispersionless limits of the quasi-continuum models, which can be used to approximate the numerically observed rarefaction waves of the discrete modified KdV equation. A systematic numerical comparison of these theoretical findings with their associated numerical counterparts finally demonstrates the good performance of the proposed quasi-continuum models in approximating these nonlinear dispersive wave patterns.