Advances in Differential Equations

Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability

Simão Correia and Mário Figueira

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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.

Article information

Source
Adv. Differential Equations, Volume 24, Number 1/2 (2019), 1-30.

Dates
First available in Project Euclid: 11 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ade/1544497233

Mathematical Reviews number (MathSciNet)
MR3910029

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35B35: Stability 35B06: Symmetries, invariants, etc. 35E99: None of the above, but in this section

Citation

Correia, Simão; Figueira, Mário. Some $L^\infty$ solutions of the hyperbolic nonlinear Schrödinger equation and their stability. Adv. Differential Equations 24 (2019), no. 1/2, 1--30. https://projecteuclid.org/euclid.ade/1544497233


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