Energy transfer model for the derivative nonlinear Schrödinger equations on the torus

Hideo Takaoka

doi:10.3934/dcds.2017253

Article Contents

2017, Volume 37, Issue 11: 5819-5841. Doi: 10.3934/dcds.2017253

This issue Previous Article Mixed elliptic problems involving the

$p-$

Laplacian with nonhomogeneous boundary conditions Next Article Estimating the fractal dimension of sets determined by nonergodic parameters

Energy transfer model for the derivative nonlinear Schrödinger equations on the torus

Hideo Takaoka^,

Department of Mathematics, Kobe University, 1-1, Rokkodai, Nada-ku, Kobe 657-8501, Japan

* Dedicated to Professor Yoshio Tsutsumi on his 60th birthday
* Dedicated to Professor Yoshio Tsutsumi on his 60th birthday

Received: August 31, 2016

Revised: May 31, 2017

Published: October 2017

The author is supported by supported by JSPS KAKENHI Grant Number 10322794

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Abstract

We consider the nonlinear derivative Schrödinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties of the particular solution consisting of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. The proof is based on the analysis of resonant dynamics via a finite dimensional ordinary differential system.

Keywords:

Nonlinear Schrödinger equations,
energy transfer.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 42B37.

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References

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