On the asymptotic behaviour as t→∞ of the solutions of the equation u t =u xxx +F[u]

Abstract

A study is made of the asymptotic behaviour of the solutions of the nonlinear nonintegrable equations of KdV type. The formal solutions in three regions: x/t>0, x/t<0, x/t^1/3=O(1) are constructed. As a result of the the investigation of the rearrangement of those solutions formulas connecting the parameters of asymptotics in different regions are obtained. The final formula corresponds the relation connecting the amplitude of the solutions which is exponentially decreasing as x→−∞ and the amplitude of the solution of the dispersion type for the KdV equation. All constructions are carried out within the context of formal solutions.