Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations

Abstract

We study the energy cascade problematic for some nonlinear Schrödinger equations on the torus $\T^2$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one $|u{|}^{2}u$. Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).

32 pages. Some references added/updated

References in corpus (1)

• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential