Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori

Abstract

In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (Comm. Math. Phys., 2014) for the square case, to obtain the $H^s$-instability ($s>1$) of the elliptic equilibrium $u=0$. We also provide the existence of solutions $u(t)$ with arbitrarily small $L^2$ norm which achieve a prescribed growth, say $\Vert u(T){\Vert }_{H^s}\ge K\Vert u(0){\Vert }_{H^s},K\gg 1$, within a time $T$ satisfying polynomial estimates, namely $0<T\le K^c$ for some $c>0$.