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Indexed by:Journal Papers

Date of Publication:2019-11-01

Journal:DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

Included Journals:SCIE

Volume:39

Issue:11

Page Number:6599-6630

ISSN No.:1078-0947

Key Words:KAM theory; almost periodic solution; Gevrey space; nonlinear Schrodinger equation

Abstract:In this paper, we study the following nonlinear Schrodinger equation
   root-1u(t) - u(xx) + V * u + epsilon f (x)vertical bar u vertical bar(4)u = 0, x is an element of T = R/2 pi Z, (1)
   where V * is the Fourier multiplier defined by <((V * u))over cap>(n) = V-n(u) over cap (n), V-n is an element of [-1, 1] and f (x) is Gevrey smooth. It is shown that for 0 <= vertical bar epsilon vertical bar << 1, there is some (V-n)(n is an element of Z) such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] to the case that the nonlinear perturbation depends explicitly on the space variable x. The main difficulty here is the absence of zero momentum of the equation.