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Finite speed of disturbance for the nonlinear Schrödinger equation

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Simão Correia
Simão Correia*
Affiliation:
CMAF-CIO and FCUL, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal (sfcorreia@fc.ul.pt)
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Abstract

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We consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

Keywords

nonlinear Schrödinger equationglobal well-posednessfinite speed of disturbance

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35A01: Existence problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1405 - 1419
Copyright
Copyright © Royal Society of Edinburgh 2018 

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