On discrete rarefaction waves in an NLS toy model for weak turbulence

2013/07/31 by Sebastian Herr, Jeremy L. Marzuola
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Classical mechanics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Geology #Mathematics #Mechanics #Physics #Rarefaction (ecology) #Statistical physics #Turbulence #math.AP #math.CA #msc:35Q55

paper · pdf · doi:10.1512/iumj.2016.65.5815

published as Indiana Univ. Math. J. (2016), Vol. 65, No. 3, pp. 753-777 · 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

arxiv created 2015/10/27 · openalex publication_date 2016/01/01 · arxiv updated 2018/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 Schr"odinger equation on the torus.

Citations

• Behavior of a Model Dynamical System with Applications to Weak Turbulence
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential
• Discrete analogue of the Burgers equation
• Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations
• Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation