Quasi-periodic solutions in a nonlinear Schrödinger equation

doi:10.1016/j.jde.2006.07.027

Journal of Differential Equations

Volume 233, Issue 2, 15 February 2007, Pages 512-542

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Abstract

In this paper, one-dimensional (1D) nonlinear Schrödinger equation $i u_{t} - u_{x x} + m u + {| u |}^{4} u = 0$ with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer $N > 1$ , the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.

MSC

37K55

35B10

35J10

35Q40

35Q55

Keywords

Schrödinger equation

Hamiltonian systems

KAM theory

Normal form

Quasi-periodic solution

Cited by (0)

¹: The work was partially done when the first author was visiting the School of Mathematics, Georgia Institute of Technology.

²: Partially supported by NSF grant DMS0204119.