Nonlinear Scattering and Analyticity Properties of Solitons

Abstract

In this paper we consider the scattering of a soliton or solitary wave by a linear potential. By careful treatment of the radiation we show that the amount of mass and energy lost by the solitary wave during a scattering event is exponentially small for strong nonlinearities. The constant associated with this exponentially small radiation is expressed in terms of the binding energy of the soliton (solitary wave), and the analyticity properties of the potential and the soliton (solitary wave). This calculation does not use integrability in any way. In the case of a delta function potential and the cubic NLS, our results agree with the more explicit results derived by Kivshar, Gredeskul, Sanchez, and Vasquez using perturbation theory based on the inverse scattering transform. Following them, we take the limit of a continuum of well separated scatterers, and derive a closed system of ordinary differential equations. Analyzing the limiting behavior of these equations for large distance Z into the medium we find that the velocity of the soliton decays as (log(Z))-1 for a delta function potential or a potential which is meromorphic as a function of a complex variable, and more slowly than (log(Z))-1 for a potential which is an entire function of a complex variable.