Growth of Sobolev norms for the quintic NLS on $\mathbb T^2$

TL;DR

This paper demonstrates the existence of solutions to the quintic NLS on a 2D torus with Sobolev norms that grow over time, using a simplified toy model and combinatorial analysis.

Contribution

It introduces a simplified toy model for the quintic NLS on the torus and analyzes Sobolev norm growth through detailed combinatorial methods.

Findings

Existence of orbits with growing Sobolev norms

Reduction to a simplified toy model

Application of combinatorial analysis

Abstract

We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.