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**Indexed by:**Journal Papers

**First Author:**Cong, Hongzi

**Correspondence Author:**Wu, Y (reprint author), Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China.

**Co-author:**Mi, Lufang,Shi, Yunfeng,Wu, Yuan

**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.