Generation of asymptotic solitons of the nonlinear Schrödinger equation by boundary data

Abstract

This article is about the focusing nonlinear Schrödinger equation on the half-line. The initial function vanishes at infinity while boundary data are local perturbations of periodic or quasi-periodic (finite-gap) functions. We study the corresponding scattering problem for the Zakharov–Shabat compatible differential equations, the representation of the solution of the nonlinear Schrödinger equation in the quarter of the (x,t)-plane through functions, which satisfy Marchenko integral equations. We use this formalism to investigate the asymptotic behavior of the solution for large time. We prove that under certain conditions a periodic (quasi-periodic) behavior at infinity of boundary data generates an unbounded train of asymptotic solitons running away from the boundary. The asymptotics of the solution shows that boundary data with periodic behavior as time tends to infinity generates a train of such asymptotic solitons even in the case when the initial function is identically zero.