## Differential and Integral Equations

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

### Stability of standing waves for some nonlinear Schrödinger equations

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

#### 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