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 (5)

• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
• Global well-posedness of the cubic nonlinear Schrödinger equation on compact manifolds without boundary
• Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential
• First and second order approximations for a nonlinear wave equation

Cited by in corpus (4)

• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
• Turbulent cascades in a truncation of the cubic Szego equation and related systems
• Global well-posedness and long-time behavior of the fractional NLS
• On discrete rarefaction waves in a nonlinear Schrödinger equation toy model for weak turbulence