2003 Stability of standing waves for some nonlinear Schrödinger equations
J. B. McLeod, C. A. Stuart, W. C. Troy
Differential Integral Equations 16(9): 1025-1038 (2003). DOI: 10.57262/die/1356060555

Abstract

The paper concerns the monotonicity with respect to $\lambda$ of the $L^2$-norm of the branch of positive solutions of the nonlinear eigenvalue problem $$ u''(x)+ g(x, u(x)^2) \ u(x) + \lambda u(x) = 0, \ x\in {\bf R}, \ \lim\limits_{|x|\rightarrow \infty} \ u(x) = 0.$$ For the particular case $ g(x) = p(x) + s^{\sigma},$ with $\sigma > 0$ and $p(x)$ an even function, decreasing for $x > 0$ and with $p(\infty) = 0$, the main theorem implies that the $L^2$-norm decreases as we increase $\lambda$ if $\sigma \leq 2$. It is also shown that this is no longer true if $\sigma > 2$. The result has implications for the orbital stability of standing waves of the nonlinear Schrödinger equation.

Citation

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J. B. McLeod. C. A. Stuart. W. C. Troy. "Stability of standing waves for some nonlinear Schrödinger equations." Differential Integral Equations 16 (9) 1025 - 1038, 2003. https://doi.org/10.57262/die/1356060555

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1037.34022
MathSciNet: MR1989539
Digital Object Identifier: 10.57262/die/1356060555

Subjects:
Primary: 34B15
Secondary: 35B35 , 35Q55

Rights: Copyright © 2003 Khayyam Publishing, Inc.

Vol.16 • No. 9 • 2003
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