Behavior of a Model Dynamical System with Applications to Weak Turbulence

Abstract

We experimentally explore solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and random phases, which supports the conjecture that energy cascades are a generic phenomenon. We also identify stationary solutions and periodic solutions in an associated problem, and find experimental evidence of hyperbolic behavior. Many of our results rely on reframing the dynamical system using a hydrodynamic formulation.

2000 AMS Subject Classification:

• 35Q55

Keywords:

• nonlinear Schrödinger equations
• frequency cascades

Acknowledgments

J.L.M. thanks the University of Toronto for hosting him during the beginning of this work, during which time he was partially supported by an NSF postdoc fellowship. Toward the end of this research, he was supported by a combination of an IBM junior faculty development award through the University of North Carolina and a guest lectureship at Universität Bielefeld. Thanks to Laurent Thomann for interesting comments and discussions regarding an early version of this paper.

G.S. was supported by NSERC. His contribution to this work was completed under NSF PIRE grant OISE-0967140 and DOE grant DE-SC0002085.

J.E.C. is partially supported by NSERC through grant number RGPIN 250233-12.

Notes

¹Indeed, using recursive linear algebra techniques, this question has been answered affirmatively thanks to an observation of Stefan Steinerberger about the recursive nature of the sequence of N for which positive solutions exist and a clever argument from Sergei Ivanov through the Math Overflow Project at http://mathoverflow.net/questions/106816 .