Far-Field Asymptotics for Multiple-Pole Solitons in the Large-Order Limit

TL;DR

This paper analyzes the large-order asymptotic behavior of multiple-pole solitons in the focusing nonlinear Schrödinger equation, revealing four distinct solution regions with different qualitative behaviors.

Contribution

It introduces a detailed asymptotic analysis of high-order multiple-pole solitons using Riemann-Hilbert problem techniques, identifying distinct solution regimes.

Findings

Identification of four qualitatively different regions in space-time

Leading-order asymptotics computed for three regions

Demonstration of complex solution behavior in the large-order limit

Abstract

The integrable focusing nonlinear Schrodinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the space-time plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions.