## Tohoku Mathematical Journal

### Stability of a stationary solution for the Lugiato-Lefever equation

#### Abstract

We study the stability of a stationary solution for the Lugiato-Lefever equation with the periodic boundary condition in one space dimension, which is a damped and driven nonlinear Schrödinger equation introduced to model the optical cavity. In this paper, we prove the Strichartz estimates for the linear damped Schrödinger equation with potential and external forcing and investigate the stability of certain stationary solutions under the initial perturbation within the framework of $L^2$.

#### Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 651-663.

Dates
First available in Project Euclid: 6 January 2012

https://projecteuclid.org/euclid.tmj/1325886285

Digital Object Identifier
doi:10.2748/tmj/1325886285

Mathematical Reviews number (MathSciNet)
MR2872960

Zentralblatt MATH identifier
1234.35251

Subjects
Secondary: 35B35: Stability

#### Citation

Miyaji, Tomoyuki; Ohnishi, Isamu; Tsutsumi, Yoshi. Stability of a stationary solution for the Lugiato-Lefever equation. Tohoku Math. J. (2) 63 (2011), no. 4, 651--663. doi:10.2748/tmj/1325886285. https://projecteuclid.org/euclid.tmj/1325886285

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