The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data

Abstract

We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes as x → +∞ and approaches an oscillatory plane wave as x → −∞. We first develop an inverse scattering transform formalism for solutions satisfying such step-like boundary conditions. Using this formalism, we prove that there exists a global solution of the corresponding Cauchy problem and establish a representation for this solution in terms of the solution of a Riemann-Hilbert problem. By performing a steepest descent analysis of this Riemann-Hilbert problem, we identify three asymptotic sectors in the half-plane t ≥ 0 of the xtplane and derive asymptotic formulas for the solution in each of these sectors. Finally, by restricting the constructed solutions to the half-line x ≥ 0, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem.

A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions

“…A more recent progress in this direction (in the case of the Korteweg-de Vries equation) has been reported in [37,39,46] (see also [38]). Another way to show existence is to infer it from the RH problem formalism (see, e.g., [42] for the case of defocusing nonlinear Schrödinger equation), where a key point consists in establishing a solution of the associated RH problem and controlling its behavior w.r.t. the spatial parameter.…”

“…Particularly, initial value problems with initial data approaching different "backgrounds" at different spatial infinities (the so-called steplike initial data) have attracted considerable attention because they can be used as models for studying expanding, oscillatory dispersive shock waves (DSW), which are large scale, coherent excitation in dispersive systems [4,40]. Large-time evolution of step-like initial data has been studied for models of uni-directional (Korteweg-de Vries equation) wave propagation [1,36] as well as bi-directional (Nonlinear Schrödinger equation) wave propagation [5,6,12,13,19,42,50].…”

On the long-time asymptotics of the modified Camassa-Holm equation with step-like initial data

“…The proof of (a), (c) and (d) is trivial. And the proof of (b) is similar as [55] and Appendix C in [54].…”

“…for a variety of integrable systems such as the KdV equation [48,49], the focusing and defocusing NLS equations [50][51][52][53][54][55][56], the modified KdV equation [57][58][59][60][61] and Camassa-Holm equation [62] among many others. A wide range of important physical phenomena manifest themselves in the behavior of solutions of such problems for large times, e.g., collisionless and dispersive shock waves [63], rarefaction waves [56],…”

The defocusing NLS equation with nonzero background: Large-time asymptotics in the solitonless region

“…Fromm, Lenells and Quirchmayr studied the long-time asymptotics for the defocusing NLS equation with the step-like boundary condition [34] q(x, t) ∼ αe 2iβx+iωt , x → −∞, 0, x → ∞.…”