Sobolev norms explosion for the cubic NLS on irrational tori

TL;DR

This paper constructs solutions to the cubic nonlinear Schrödinger equation on irrational tori that exhibit unbounded growth in Sobolev norms, using quasi-resonances and Diophantine approximation to analyze the dynamics.

Contribution

It demonstrates Sobolev norm growth for solutions on irrational tori, leveraging quasi-resonances and Diophantine properties to understand energy transfer mechanisms.

Findings

Constructed solutions with Sobolev norm growth for all s>0, s≠1.

Established solutions with norms from arbitrarily small to large on sets with positive Hausdorff dimension.

Provided estimates on the time scale of Sobolev norm explosion.

Abstract

We consider the cubic nonlinear Schr\"odinger equation on $2$-dimensional irrational tori. We construct solutions which undergo growth of Sobolev norms. More concretely, for every $s>0$, $s\neq 1$ and almost every choice of spatial periods we construct solutions whose $H^s$ Sobolev norms grow by any prescribed factor. Moreover, for a set of spatial periods with positive Hausdorff dimension we construct solutions whose Sobolev norms go from arbitrarily small to arbitrarily large. We also provide estimates for the time needed to undergo the norm explosion. Note that the irrationality of the space periods decouples the linear resonant interactions into products of $1$-dimensional resonances, reducing considerably the complexity of the resonant dynamics usually used to construct transfer of energy solutions. However, one can provide these growth of Sobolev norms solutions by using quasi-resonances relying on Diophantine approximation properties of the space periods.