Nonlinear dispersive waves in the discrete modified KdV equation

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.

Figures (8)

• Figure 1: The evolution of the Riemann problem associated with the discrete mKdV equation \ref{['e: dmKdV equation']}. The dynamics is shown at $t = 1000$ with the backgrounds of $u_- = 0$ and $u_+ = 0.2$ for the panel $(a)$, while $u_- = 0.2$ and $u_+ = 0$ for $(b)$.
• Figure 2: The comparison of the linear dispersion relations of the three models in Eqs. \ref{['e: dmKdV equation']} (blue), \ref{['e: second continuum model']} (red) and \ref{['e: regularized model']} (magenta).
• Figure 3: The potential curves in Eq. \ref{['potential curves']}. The panels $(a)$ and $(b)$ depict the potential curve of $P_{\text{n}}(u)$ and $P_{\text{r}}(\tilde{u})$, respectively.
• Figure 4: The "box-type" initial condition with $u_l = 0$, $u_p = 1$ and $\delta = 1$. The two initial jumps are located at $x_l = -1000$ and $x_r = 1000$, respectively.
• Figure 5: The comparison of the numerical RW (discrete red dots) of the discrete mKdV equation \ref{['e: dmKdV equation']} with the self-similar solution (blue curve) of the quasi-continuum models at $t = 1000$ with $u_- = 0.2$ and $u_+ = 0$.