Stability Analysis Of The Peregrine Solution Via Squared Eigenfunctions

Abstract

A preliminary numerical investigation involving ensembles of perturbed initial data for the Peregrine soliton (the lowest order rational solution of the nonlinear Schrödinger equation) indicates that it is unstable [16]. In this paper we analytically investigate the linear stability of the Peregrine soliton, appealing to the fact that the Peregrine solution can be viewed as the singular limit of a single mode spatially periodic breathers (SPB). The "squared eigenfunction" connection between the Zakharov-Shabat (Z-S) system and the linearized NLS equation is employed in the stability analysis. Specifically, we determine the eigenfunctions of the Z-S system associated with the Peregrine soliton and construct a family of solutions of the associated linearized NLS (about the Peregrine) in terms of quadratic products of components of the eigenfunctions (i.e., the squared eigenfunction). We find there exist solutions of the linearization that grow exponentially in time, thus showing the Peregrine soliton is linearly unstable.

Publication Date

10-12-2017

Publication Title

AIP Conference Proceedings

Volume

1895

Document Type

Article; Proceedings Paper

Personal Identifier

scopus

DOI Link

https://doi.org/10.1063/1.5007371

Socpus ID

85031692886 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85031692886

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