Instability in nonlinear Schrödinger breathers

Abstract: We consider the focusing Nonlinear Schrödinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study the initial value problem for perturbations of the background...

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Autor principal:	Muñoz,Claudio
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2017
Materias:	modulation instability well-posedness Schrödinger,Peregrine breather
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000400653
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spelling	oai:scielo:S0716-091720170004006532018-01-08Instability in nonlinear Schrödinger breathersMuñoz,Claudio modulation instability well-posedness Schrödinger,Peregrine breather Abstract: We consider the focusing Nonlinear Schrödinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study the initial value problem for perturbations of the background wave in Sobolev spaces. It is well-known that the associated linear dynamics for this problem describes a phenomenon known in the literature as modulational instability, also recently related to the emergence of rogue waves in ocean dynamics. In qualitative terms, small perturbations of the background state increase its size exponentially in time. In this paper we show that, even if there is no time decay for the linear dynamics due to the modulationally unstable regime, the equation is still locally well-posed in H s, s > . We apply this result to give a rigorous proof of the unstable character of two well-known NLS solutions: the Peregrine and Kuznetsov-Mabreathers.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.36 n.4 20172017-12-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000400653en10.4067/S0716-09172017000400653
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	modulation instability well-posedness Schrödinger,Peregrine breather
spellingShingle	modulation instability well-posedness Schrödinger,Peregrine breather Muñoz,Claudio Instability in nonlinear Schrödinger breathers
description	Abstract: We consider the focusing Nonlinear Schrödinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study the initial value problem for perturbations of the background wave in Sobolev spaces. It is well-known that the associated linear dynamics for this problem describes a phenomenon known in the literature as modulational instability, also recently related to the emergence of rogue waves in ocean dynamics. In qualitative terms, small perturbations of the background state increase its size exponentially in time. In this paper we show that, even if there is no time decay for the linear dynamics due to the modulationally unstable regime, the equation is still locally well-posed in H s, s > . We apply this result to give a rigorous proof of the unstable character of two well-known NLS solutions: the Peregrine and Kuznetsov-Mabreathers.
author	Muñoz,Claudio
author_facet	Muñoz,Claudio
author_sort	Muñoz,Claudio
title	Instability in nonlinear Schrödinger breathers
title_short	Instability in nonlinear Schrödinger breathers
title_full	Instability in nonlinear Schrödinger breathers
title_fullStr	Instability in nonlinear Schrödinger breathers
title_full_unstemmed	Instability in nonlinear Schrödinger breathers
title_sort	instability in nonlinear schrödinger breathers
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2017
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000400653
work_keys_str_mv	AT munozclaudio instabilityinnonlinearschrodingerbreathers
_version_	1718439829044199424

Instability in nonlinear Schrödinger breathers

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