The CamassaHolm equation as the longwave limit of the improved Boussinsq equation and of a class of nonlocal wave equations
Erbay, Hüsnü Ata and Erbay, Saadet and Erkip, Albert (2015) The CamassaHolm equation as the longwave limit of the improved Boussinsq equation and of a class of nonlocal wave equations. (Accepted/In Press) AbstractIn the present study we prove rigorously that in the longwave 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 CamassaHolm 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 CamassaHolm 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 ϵ and δ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the BenjaminBonaMahony approximation and the Kortewegde Vries approximation Available Versions of this Item The CamassaHolm equation as the longwave limit of the improved Boussinsq equation and of a class of nonlocal wave equations. (deposited 25 Jan 2016 10:51) [Currently Displayed]
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