Strong convergence on space-time branches for the multi-soliton asymptotics

Establish strong convergence, on suitable space-time branches, for the modulated multi-soliton decomposition of solutions to the one-dimensional defocusing nonlinear Schrödinger equation i∂tΨ + ∂x^2Ψ + Ψ f(|Ψ|^2) = 0 with non-vanishing boundary condition |Ψ| → 1 as |x| → ∞: specifically, determine constants σ_- < σ_+ such that the asymptotic convergences towards the traveling waves described in Theorem (asymptotic stability of a chain) hold as strong convergences when restricted to the spatial window σ_- t ≤ x ≤ σ_+ t.

Background

The paper proves asymptotic stability of well-prepared chains of traveling waves and constructs asymptotic N-soliton-like solutions for the defocusing NLS with general nonlinearity and non-zero boundary conditions. The convergence obtained in Theorem (asymptotic stability of a chain) is in a weak sense adapted to the energy framework and cannot, in general, be strengthened to strong convergence globally due to phenomena such as smaller soliton emergence and phase winding.

The authors note, however, that stronger convergence might be achievable in refined settings by restricting attention to suitable space-time branches between characteristic speeds. They formulate a specific conjecture identifying the existence of constants defining such branches on which strong convergence should hold.