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Numerical study of the Amick-Schonbek system

C. Klein, J. -C. Saut

Abstract

The aim of this paper is to present a survey and a detailed numerical study on a remarkable Boussinesq system describing weakly nonlinear, long surface water waves. In the one-dimensional case, this system can be viewed as a dispersive perturbation of the hyperbolic Saint-Venant (shallow water) system. The asymptotic stability of the solitary waves is numerically established. Blow-up of solutions for initial data not satisfying the non-cavitation condition as well as the appearence of dispersive shock waves are studied.

Numerical study of the Amick-Schonbek system

Abstract

The aim of this paper is to present a survey and a detailed numerical study on a remarkable Boussinesq system describing weakly nonlinear, long surface water waves. In the one-dimensional case, this system can be viewed as a dispersive perturbation of the hyperbolic Saint-Venant (shallow water) system. The asymptotic stability of the solitary waves is numerically established. Blow-up of solutions for initial data not satisfying the non-cavitation condition as well as the appearence of dispersive shock waves are studied.
Paper Structure (9 sections, 1 theorem, 23 equations, 23 figures)

This paper contains 9 sections, 1 theorem, 23 equations, 23 figures.

Table of Contents

  1. Introduction
  2. The Cauchy problem
  3. The two-dimensional case
  4. Solitary waves of the Amick-Schonbek system
  5. Numerical study of perturbed solitary waves
  6. Localised initial data
  7. Non-cavitation initial data
  8. Dispersive shock waves
  9. Conclusion

Key Result

Theorem 2.1

Let $s>1/2.$ For $(\zeta_,u_0)\in H^s({\mathbb R})\times H^{s+1}({\mathbb R})$ such that $1+\zeta_0>0,$ the Boussinesq system AS has a solution $(\zeta,u)\in C({\mathbb R}_+,H^s({\mathbb R})\times H^{s+1}({\mathbb R}))\cap C^1({\mathbb R}_+,H^{s-1}({\mathbb R})\times H^s({\mathbb R})).$ This solutio

Figures (23)

  • Figure 1: Solitary waves of the Amick-Schonbeck system for different values of the velocity $C=2,1.8,1.6,1.4,1.2,1.1$ from top to bottom, on the left $V$, on the right $Q$.
  • Figure 2: Solitary waves of the Amick-Schonbeck system for different values of the velocity $C=3, 2.8, 2.6, 2.4, 2.2$ from top to bottom, on the left $V$, on the right $Q$.
  • Figure 3: The $L^{\infty}$ norm of the functions $Q$ of Fig. \ref{['figASsolc23']}.
  • Figure 4: Solution to the Amick-Schonbeck system (\ref{['AS']}) for the initial data $\eta(x,0) = 1.1 Q_{2}(x)$, $v(x,0) = V_{2}(x)$ for $t=10$, on the left $\eta$, on the right $v$.
  • Figure 5: $L^{\infty}$ norms of the solution to the Amick-Schonbeck system (\ref{['AS']}) for the initial data $\eta(x,0) = 1.1 Q_{2}(x)$, $v(x,0) = V_{2}(x)$ in dependence of time, on the left for $\eta$, on the right for $v$.
  • ...and 18 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8