Solitary wave evolution for mKdV equations

Abstract

The evolution of an initial condition into soliton(s) is the classic problem for the Korteweg-de Vries (KdV) equation. While this evolution is theoretically given by the inverse scattering solution of the KdV equation, in practice only the final steady state can be easily obtained from inverse scattering. However, an approximate method based on the conservation laws for the KdV equation has been found to give very accurately the evolution of an initial condition into soliton(s). This approximate method Also gives a criterion for the number of solitons formed. In the present work, this method is extended to describe the evolution of an initial condition into solitary wave(s) for mKdV equations, these equations having the same dispersive term as the KdV equation, but a nonlinear term of the form u n u x , where n ≥ 1 is a positive integer. It is found that for n < 4, the behaviour of the mKdV equation is similar to the KdV equation in that solitary wave(s) evolve from an arbitrary initial condition. However, for n ≥ 4, it is found that an initial condition of sufficiently small amplitude decays into dispersive radiation with no solitary wave being formed. For an initial amplitude exceeding a threshold, it is found that the amplitude blows-up. The solutions of the approximate equations are compared with full numerical solutions of the mKdV equation and good agreement is found.