The Camassa-Holm approximation to the double dispersion equation for arbitrarily long times

Abstract

In the present paper we prove the validity of the Camassa-Holm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an infinite elastic medium. First we show formally that the right-going wave solutions of the double dispersion equation can be approximated by the solutions of the Camassa-Holm equation in the long wave limit. Then we rigorously prove that the solutions of the double dispersion and the Camassa-Holm equations remain close over a long time interval, determined by two small parameters measuring the effects of nonlinearity and dispersion.