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

Comparison of nonlocal nonlinear wave equations in the long-wave limit

“…For a comprehensive treatment of the methods on comparisons of solutions between those unidirectional equations and the parent equations (for instance, in the context of a shallow water approximation) we refer to [12][13][14] for the case of the KDV and the BBM equations and to [4,7] for the case of the CH equation (see, for instance, [5,15] for comparisons of two-dimensional model equations). In a recent study [16], the present authors have made similar comparisons between (1) in the context of nonlocal elasticity and the CH equation. For emphasis, we remind the reader that all these studies consider unidirectional approximations of nonlinear dispersive equations whereas the present work is about bidirectional approximations.…”

Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations

“…The first assumption is about the evolution operator generated by L. The following discussion shows that this assumption is valid. K is a bounded operator on H s due to (5). Thus KD x maps H s+1 into H s and so L : X s+1 → X s is a bounded operator independent of τ .…”

“…The long-time existence results of the studies focused on water waves [8,9,10] have been also used for the rigorous justification of approximate asymptotical models (such as the Green-Naghdi equations, the shallow water equations and the Boussinesq system) starting from the Euler equations describing the motion of an inviscid, incompressible fluid. In addition to the studies about asymptotic models of water waves, there are also studies presenting the rigorous derivation of various asymptotic models for nonlinear elastic waves in the long-wave-small-amplitude regime (for instance we refer the reader to [5] where the Camassa-Holm equation and (1) are compared). The present research is motivated by the long-time existence results that were reported for water waves and aims to extend those results to elastic waves propagating in nonlocal elastic solids.…”

On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations

“…In (1), u = u(x, t) is a real-valued function, and ǫ and δ are two small positive parameters measuring the effects of nonlinearity and dispersion, respectively. In [11], by a proper choice of initial data, we restricted our attention to the right-going solutions of the IB equation and showed that, for small amplitude long waves, they are well approximated by associated solutions of a single CH equation [4]. In the present study we remove the assumption about the solutions being unidirectional and consider solutions traveling in both directions with general initial disturbances.…”

“…which appears as a relevant model in various areas of physics (see, e.g. [15,18,8] for solid mechanics), and we proceed along our analysis of the Camassa-Holm (CH) approximation of the IB equation initiated in [11]. In (1), u = u(x, t) is a real-valued function, and ǫ and δ are two small positive parameters measuring the effects of nonlinearity and dispersion, respectively.…”