A remark on linearly unstable standing wave solutions to NLS

✍ Scribed by Tetsu Mizumachi

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 226 KB
• Volume: 64
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.05.043

✦ Synopsis

We obtain an optimal growth estimate of a semigroup generated by a linearized operator around a standing wave solution nonlinear Schrödinger equations in two-dimension. Using the growth estimate of the semigroup, we prove that a linearly unstable standing wave solution is orbitally unstable and that instability of the standing wave solution is mainly caused by a mode of an eigenfunction associated with the rightmost (or the leftmost) eigenvalues of the linearized operator. Our result is obtained by using the method of Yajima and Cuccagna that proved L p -boundedness of the wave operator.