Differential and Integral Equations

Stability of standing waves for some nonlinear Schrödinger equations

J. B. McLeod, C. A. Stuart, and W. C. Troy

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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.

Article information

Source
Differential Integral Equations, Volume 16, Number 9 (2003), 1025-1038.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060555

Mathematical Reviews number (MathSciNet)
MR1989539

Zentralblatt MATH identifier
1037.34022

Subjects
Primary: 34B15: Nonlinear boundary value problems
Secondary: 35B35: Stability 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

McLeod, J. B.; Stuart, C. A.; Troy, W. C. Stability of standing waves for some nonlinear Schrödinger equations. Differential Integral Equations 16 (2003), no. 9, 1025--1038. https://projecteuclid.org/euclid.die/1356060555


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