Nonlinear smoothing for the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity

TL;DR

This paper investigates the nonlinear smoothing effects and local well-posedness of periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearities, showing solutions become smoother than initial data and establishing new well-posedness results.

Contribution

It introduces novel nonlinear smoothing properties and improves local well-posedness results for these equations and related fifth-order KdV equations.

Findings

Solutions exhibit nonlinear smoothing beyond initial regularity

Established local well-posedness for large dispersion

Improved results for non-integrable fifth-order KdV equations

Abstract

We consider the periodic dispersion generalized Benjamin-Ono equations with polynomial nonlinearity. We establish the nonlinear smoothing properties of these equations, according to which the difference between the solution and the linear evolution is smoother than the initial data. In addition, we establish new local well-posedness results for these equations when the dispersion is sufficiently large. Our method also improves known local well-posedness results for a class of non-integrable fifth-order KdV equations.