Growth Bound and Nonlinear Smoothing for the Periodic Derivative Nonlinear Schr\"odinger Equation

TL;DR

This paper proves polynomial growth bounds for solutions to the periodic derivative nonlinear Schrödinger equation and reveals a nonlinear smoothing effect that enhances regularity beyond initial data.

Contribution

It introduces a nonlinear smoothing effect for a gauge-transformed version of the equation, leading to polynomial growth bounds for solutions.

Findings

Polynomial-in-time growth bounds for solutions.

Existence of a nonlinear smoothing effect.

Higher regularity of solutions after gauge transformation.

Abstract

A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb T)$ solutions to the derivative nonlinear Schr\"odinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.