The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

Abstract

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.

24 pages, to appear in Discrete and Continuous Dynamical Systems

References in corpus (3)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Variational derivation of the Camassa-Holm shallow water equation
• Derivation of the Camassa-Holm equations for elastic waves

Cited by in corpus (1)

• Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations