On the growth of high Sobolev norms for certain one-dimensional Hamiltonian PDEs

TL;DR

This paper investigates the long-term growth of Sobolev norms for solutions to fractional cubic Schrödinger equations on the torus, demonstrating polynomial growth bounds for most cases and analyzing specific half-wave scenarios.

Contribution

It establishes polynomial bounds on Sobolev norm growth for a class of fractional Schrödinger equations, extending understanding beyond the well-studied cases.

Findings

Solutions grow at most polynomially in time for α ≠ 1

The case α=1 (half-wave) exhibits no dispersive property

Analysis of cubic and quadratic half-wave equations

Abstract

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{\"o}dinger equation on the torus :$$i \partial\_t u = |D|^\alpha u+|u|^2 u, \quad u(0, \cdot)=u\_0,$$where $\alpha$ is a real parameter. We show that, apart from the case $\alpha = 1$, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.