Energy cascades for NLS on the torus

Abstract

We consider the nonlinear Schrodinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.

15 pages, 2 figures

References in corpus (3)

• Birkhoff Normal Form and Hamiltonian PDEs
• The quasi-periodic doubling cascade in the transition to weak turbulence
• Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities

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• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
• Three-dimensional gravity and instability of ${\text{AdS}}_3$
• Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
• 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
• Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori
• Growth of Sobolev norms for the analytic NLS on ${\mathbb{T}}^2$
• Stability of the Cubic Nonlinear Schrodinger Equation on Irrational Tori
• Modified scattering for the cubic Schrödinger equation on Diophantine waveguides