A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

Abstract

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary‐order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher‐order “rogue wave” solutions in an inverse‐scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.

Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

“…Meanwhile, the classical inverse scattering method on plane wave background can not involve the rogue waves and high order rogue waves directly; we need to perform a sensitive limiting procedure to obtain them. Recently, Bilman and Miller [8] proposed a new scheme called robust inverse scattering transform to deal with the spectral singularity with the aid of the Darboux transformation. Meanwhile, it is hopeful to be used in analyzing the long-time asymptotics for the rogue waves and high order rogue wave on the plane wave background.…”

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

“…We now introduce the basic Riemann-Hilbert problem used to define the multipole solitons we study. This representation was derived in [1] using the recently introduced robust inverse-scattering transform [3].…”

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

“…The finite-gap method was employed to relate the unstable modes on the constant background with the occurrence of rogue waves [24,25]. Rogue waves of infinite order were constructed in [10] based on recent developments in the inverse scattering method [9]. Rogue waves of the soliton superposition were studied asymptotically in the limit of many solitons [8,30].…”