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.

arXiv comment33 pages, 14 figures

References in corpus (7)

• Dispersive shock waves and modulation theory
• Multi-soliton, multi-breather and higher-order rogue wave solutions to the complex short pulse equation
• Exact solutions to the focusing nonlinear Schrodinger equation
• Exact Solutions to the Sine-Gordon Equation
• Large-degree asymptotics of rational Painleve-II functions. II
• Influence of small dispersion on self-focusing in spatially one-dimensional case
• Semiclassical soliton ensembles for the three-wave resonant interaction equations

Cited by in corpus (12)

• Multi-breathers and high order rogue waves for the nonlinear Schrödinger equation on the elliptic function background
• Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy
• Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background
• Rogue waves on the double-periodic background in the focusing nonlinear Schrodinger equation
• Far-Field Asymptotics for Multiple-Pole Solitons in the Large-Order Limit
• Darboux transformation and solitonic solution to the coupled complex short pulse equation
• Rigorous asymptotics of a KdV soliton gas
• Asymptotic analysis of high order soliton for the Hirota equation
• Soliton shielding of the focusing Nonlinear Schrödinger Equation
• Broader Universality of Rogue Waves of Infinite Order
• Properties of synchronous collisions of solitons in the Korteweg - de Vries equation
• Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type