Establishing Quantitative Understanding of Energy Transfer to High Frequency in Nonlinear Dispersive Equations

Abstract

We present a family of particular solutions to a Hamiltonian system which was derived to study energy transfer to higher Fourier modes in solutions to the cubic defocusing nonlinear Schrödinger equation. The solutions in this family are not direct solutions to this nonlinear Schrödinger equation, but instead approximate solutions which transfer energy to higher Fourier modes. Our numerical work follows and expands upon work done in [4] and [8], where the existence of solutions exhibiting these properties was proven non-constructively. The solutions presented here depend heavily upon phase interactions in the Hamiltonian system, which has previously been poorly understood.