Soliton caustics

Abstract

Analysis of wave propagation in nonlinear inhomogeneous media gives rise to fundamental soliton-bearing equations, such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NS) equations, with coefficients depending slowly upon an evolution variable. In the present paper we analyze the passage of a KdV and an NS soliton through critical points where a coefficient in front of a nonlinear or dispersive term in the corresponding equation changes its sign. Additional terms describing, respectively, higher nonlinearity or higher dispersion are taken into account. For the case when the nonlinear coefficient changes its sign in the KdV equation, we demonstrate that, within the framework of the corresponding modified kdV equation, a primordial soliton passing the critical point transforms into a train of secondary solitons of the opposite polarity. Earlier, this effect was observed in numerical simulations. In the case when the dispersion coefficient of the KdV equation changes the sign, the effect is more trivial: the primordial soliton decays into quasilinear dispersive waves. The same effect takes place when the nonlinear coefficient changes its sign in the NS equation. In the case when this happens with the dispersion coefficient in the NS equation, the soliton's amplitude grows, and eventually the soliton breaks down. In this case, numerical simulation seems to be the only way to describe in detail the breakdown of the soliton at the ultimate stage. At last, we demonstrate that the change of sign of the nonlinear coefficient in the Burgers equation entails decay of a corresponding shock wave. Implementation of the soliton caustics in dynamics of ocean waves is discussed in some detail.