Details

Autor(en) / Beteiligte

• Bilman, Deniz
• Buckingham, Robert

Titel

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

Ist Teil von

• Journal of nonlinear science, 2019-10, Vol.29 (5), p.2185-2229

Ort / Verlag

New York: Springer US

Erscheinungsjahr

2019

Link zum Volltext

Quelle

SpringerLink (Online service)

Beschreibungen/Notizen

• 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 2 n . 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.