On discrete rarefaction waves in a nonlinear Schrödinger equation toy model for weak turbulence

Abstract

We explore further the rarefaction wave-like solutions recently discussed in work of the second author with J. Colliander, T. Oh and G. Simpson for a model Hamiltonian dynamical system derived by Colliander-Keel-Staffilani-Takaoka-Tao to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus.

24 pages, 5 figures. Contains a brief summary of results from the second author's work with J. Colliander, T. Oh and G. Simpson in arXiv:1209.0827. Improved exposition, fixed typos and added a Numerical Section relating to a connection to the Toda Lattice

References in corpus (6)

• Resonant dynamics for the quintic non linear Schrödinger equation
• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
• Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential
• Discrete Analog of the Burgers Equation

Cited by in corpus (2)

• Standing and Traveling Waves in a Nonlinearly Dispersive Lattice Model
• Thermalization in a nonlinear variant of the discrete nonlinear Schrödinger Equation