Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

Nan Lu

doi:10.3934/dcds.2015.35.3533

Article Contents

2015, Volume 35, Issue 8: 3533-3567. Doi: 10.3934/dcds.2015.35.3533

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Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

Nan Lu^1,

1.
14 East Packer Avenue, Christmas-Saucon Hall, Lehigh University, Bethlehem, PA 18015

Received: March 31, 2014

Revised: December 31, 2014

Published: May 2017

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Abstract

We construct two families of non-localized standing waves for the hyperbolic cubic nonlinear Schrödinger equation \[iu_t+u_{xx}-u_{yy}+|u|^2u=0.\] The first family of standing waves consists of solutions which correspond to some generalized breathers for each fixed time $t$, while solutions in the second family are periodic both in $x$ and $y$. The second family of solutions were numerically observed by Vuillon, Dutykh and Fedele in a recent preprint [17].

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Mathematics Subject Classification: Primary: 37C27, 37C29; Secondary: 35B25, 35B36.

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