Tohoku Mathematical Journal
2011
December
SECOND SERIES VOL. 63, NO. 4

Tohoku Math. J.
63 (2011), 651-663

Title STABILITY OF A STATIONARY SOLUTION FOR THE LUGIATO-LEFEVER EQUATION

Author Tomoyuki Miyaji, Isamu Ohnishi and Yoshi Tsutsumi

(Received January 26, 2011, revised May 6, 2011)
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 $L^2$.

2000 Mathematics Subject Classification. Primary 35Q55; Secondary 35B35.

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