Normal Form and Solitons in the Shallow Water Waves

Abstract

It is well known that the shallow water waves can be described by the KdV equation in the first order approximation of an asymptotic perturbation expansion. The KdV equation admits a family of solitary waves called solitons having the remarkable property that the result of interaction of solitons leaves their shape unaltered, except a phase shift. This elastic interaction is due to the fact that the KdV equation possesses an infinite number of conserved quantities. (This is often to be a definition of the integrability.) However, for the problem related to large time behavior of the solutions or large amplitude solitary waves, one needs to study the effects of the higher order terms which are neglected in the derivation of the KdV equation.