Growth of Sobolev norms in the cubic defocusing nonlinear Schr\"odinger equation with a convolution potential

Abstract

Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} the existence of solutions of the cubic defocusing nonlinear Schr\"odinger equation in the two torus with $s$-Sobolev norm growing in time. In this paper we generalize their result to the cubic defocusing nonlinear Schr\"odinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schr\"odinger equation in \cite{GuardiaK12}. The results we obtain can deal with any potential in $H^{s_0}(\TT^2)$, $s_0>0$.