Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Abstract

We analyze the large-$n$ behavior of soliton solutions of the integrable focusing nonlinear Schrödinger equation with associated spectral data consisting of a single pair of conjugate poles of order $2n$. Starting from the zero background, we generate multiple-pole solitons by $n$-fold application of Darboux transformations. The resulting functions are encoded in a Riemann–Hilbert problem using the robust inverse-scattering transform method recently introduced by Bilman and Miller. For moderate values of n we solve the Riemann–Hilbert problem exactly. With appropriate scaling, the resulting plots of exact solutions reveal semiclassical-type behavior, including regions with high-frequency modulated waves and quiescent regions. We compute the boundary of the quiescent regions exactly and use the nonlinear steepest-descent method to prove the asymptotic limit of the solitons is zero in these regions. Finally, we study the behavior of the solitons in a scaled neighborhood of the central peak with amplitude proportional to $n$. We prove that in a local scaling the solitons converge to functions satisfying the second member of the Painlevé-III hierarchy in the sense of Sakka. This function is a generalization of a function recently identified by Suleimanov in the context of geometric optics and by Bilman, Ling, and Miller in the context of rogue-wave solutions to the focusing nonlinear Schrödinger equation.

Publication
Journal of Nonlinear Science, 29, 2185–2229, 2019
Deniz Bilman
Deniz Bilman
Assistant Professor

Assistant Professor of Mathematics. Research interests include nonlinear waves, Riemann-Hilbert problems.

Robert Buckingham
University of Cincinnati

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