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 Lp-boundedness of the wave operator. © 2005 Elsevier Ltd. All rights reserved.
Mizumachi, T. (2006). A remark on linearly unstable standing wave solutions to NLS. Nonlinear Analysis, Theory, Methods and Applications, 64(4), 657–676. https://doi.org/10.1016/j.na.2005.05.043