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January/February 2019 Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability
Simão Correia, Mário Figueira
Adv. Differential Equations 24(1/2): 1-30 (January/February 2019).

Abstract

Consider the hyperbolic nonlinear Schrödinger equation $\mathrm {(HNLS)}$ over $\mathbb R^d$ $$ iu_t + u_{xx} - \Delta_{\textbf{y}} u + \lambda |u|^\sigma u=0. $$ We deduce the conservation laws associated with $\mathrm {(HNLS)}$ and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including hyperbolically symmetric solutions, spatial plane waves and spatial plane waves, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a stability result for both spatial plane waves and spatial standing waves with respect to small $H^1$ perturbations.

Citation

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Simão Correia. Mário Figueira. "Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability." Adv. Differential Equations 24 (1/2) 1 - 30, January/February 2019.

Information

Published: January/February 2019
First available in Project Euclid: 11 December 2018

zbMATH: 07192791
MathSciNet: MR3910029

Subjects:
Primary: 35B06, 35B35, 35E99, 35Q55

Rights: Copyright © 2019 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.24 • No. 1/2 • January/February 2019
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