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This paper has been submitted for publication in Annals of Partial Differential Equations. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1812625 and DMS-2204896 (Miller). Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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General rogue waves of infinite order: exact properties, asymptotic behavior, and effective numerical computation

Deniz Bilman and Peter D. Miller

DB: Department of Mathematical Sciences, University of Cincinnati
PDM: Department of Mathematics, University of Michigan, Ann Arbor

Abstract:

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schroödinger equations. We establish the following key property of these solutions: they are all in \(L^2(\mathbb{R})\) with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

The solution \(\Psi(X, T; \mathbf{G}, B)\) computed with RogueWaveInfiniteNLS.jl with \(a = b = B = 1\). RogueWaveInfiniteNLS.jl is a software package developed in this work for the Julia programming language to compute rogue waves of infinite order through numerical solution of suitable Riemann-Hilbert problems.