A NEKHOROSHEV-TYPE THEOREM FOR THE NONLINEAR SCHRODINGER EQUATION ON THE TORUS

We prove a Nekhoroshev type theorem for the nonlinear Schrodinger equation iu(t) = -Delta u + V star u + partial derivative((u) over barg)(u,(u) over bar), x is an element of T-d, where V is a typical smooth Fourier multiplier and g is analytic in both variables. More precisely, we prove that if the initial datum is analytic in a strip of width rho > 0 whose norm on this strip is equal to epsilon, then if epsilon is small enough, the solution of the nonlinear Schrodinger equation above remains analytic in a strip of width rho/2, with norm bounded on this strip by C epsilon over a very long time interval of order epsilon(-sigma)|ln epsilon|(beta), where 0 < beta < 1 is arbitrary and C > 0 and sigma > 0 are positive constants depending on beta and rho.