Dynamics of Partial Differential Equations

Volume 10 (2013)

Number 2

Dynamics on resonant clusters for the quintic non linear Schrödinger equation

Pages: 157 – 169

DOI: http://dx.doi.org/10.4310/DPDE.2013.v10.n2.a2

Authors

Emanuele Haus (Laboratoire de Mathématiques J. Leray, Université de Nantes, France)

Laurent Thomann (Laboratoire de Mathématiques J. Leray, Université de Nantes, France)

Abstract

We construct solutions to the quintic nonlinear Schrödinger equation on the circle$$ i\partial_t u+\partial_{x}^{2}u = \nu \ |u|^4u,\quad \nu\ll1, \ x\in \mathbb{S}^{1},\ t\in \mathbb{R},$$with initial conditions supported on arbitrarily many different resonant clusters. This is a sequel to the work [5] of Benoît Grébert and the second author.

Keywords

nonlinear Schrödinger equation, Resonant normal form, energy exchange

2010 Mathematics Subject Classification

35B34, 35B35, 35Q55, 37K45

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