Long Time Stability Of Plane Wave Solutions To Schr?dinger Equation On Torus

We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrodinger equation(NLS)in the defocusing(?=1)or focusing(?=-1)case,(?).More precisely,in a Gevrey space (?) for some positive constant ?,we show that solution with the initial datum in the 4?-neighborhood of the plane wave solution still stays in the C?-neighborhood(C>4)of the plane wave solution for a subexponential long time |t|??-?|ln?|g,where?=min{1/4,?-?'},?>?'>0 and 0<g<1/6.