Numerical study of the Amick-Schonbek system in 2D

TL;DR

The paper numerically investigates the 2D Amick-Schonbek Boussinesq system, addressing key questions about long-wave dynamics in the Boussinesq regime. It employs a pseudospectral 2D solver on a large torus with RK4 time stepping to study line solitary waves, the non-cavitation condition, localized data, and dispersive shock waves. The main findings demonstrate transverse stability of line solitary waves, no emergence of stable 2D lump solutions, and cusp formation when the non-cavitation constraint is violated, with dispersive effects producing radiation and oscillatory structures in the near-singularity regime. These results provide numerical evidence toward understanding the global behavior of the 2D Amick-Schonbek system and guide future theoretical work and comparisons with related shallow-water models such as Serre–Green–Naghdi equations.

Abstract

A numerical study of the 2D Amick-Schonbek Boussinesq system is presented. Numerical evidence is given for the transverse stability of the 1D solitary waves that are line solitary waves of the 2D equations. It is shown that initial data not satisfying the non-cavitation condition can lead to the formation of a gradient catastrophe in finite time. The numerical propagation of localised smooth initial data does not lead to the formation of stable structures localised in both spatial directions. For initial data satisfying the non-cavitation condition, smooth solutions appear to exist for all times.

Figures (29)

• Figure 1: Solution to the AS system (\ref{['2D']}) for the initial data (\ref{['c2gauss']}) for the $+$ sign, on the left $\eta$ for $t=0$, on the right for $t=20$.
• Figure 2: Solution to the AS system (\ref{['2D']}) for the initial data (\ref{['c2gauss']}) for $t=20$, on the left $v_{x}$, on the right $v_{y}$.
• Figure 3: $L^{\infty}$ norm of the solution $\eta$ to the AS system (\ref{['2D']}) for the initial data (\ref{['c2gauss']}), on the left for the $+$ sign, on the right for the $-$ sign.
• Figure 4: Solution to the AS system (\ref{['2D']}) for the initial data (\ref{['c2gaussvx']}), in the upper row on the left $v_{x}$ for $t=0$, on the right for $t=20$, in the lower row on the left $\eta$ and on the right $v_{y}$, both for t=20.
• Figure 5: $L^{\infty}$ norms of solutions $\eta$ to the AS system (\ref{['2D']}) for various initial data, on the left for the solution shown in Fig. \ref{['figc2gaussvx']}, in the middle for the solution shown in Fig. \ref{['figc2gaussvy']}, on the right for the solution shown in Fig. \ref{['figc2cosv']}.

Theorems & Definitions (4)

• Remark 1.1
• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3