Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions
Stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under nonlinear perturbations is investigated. Using the Mourre estimate and the Fermi Golden rule for non-elliptic and nonlinear equations the author shows that for the wave equation such solutions are unstable for generic perturbations and gives spectral conditions which guarantee instability. In this case of the Schrödinger equation periodic solutions are stable while quasiperiodic ones are not. These results are extended to periodic solutions of nonlinear equations. Some illustrative examples are considered.
scientific article; zbMATH DE number 221163