Unconditional uniqueness for the periodic modified Benjamin-Ono equation by normal form approach

TL;DR

This paper proves the unconditional uniqueness of solutions to the periodic modified Benjamin-Ono equation in a certain function space using a normal form approach with infinite reduction steps.

Contribution

It introduces a novel normal form method with infinite reduction steps to establish uniqueness in the periodic setting for the modified Benjamin-Ono equation.

Findings

Uniqueness holds in $L^ abla_t(H^s( abla))$ for $s>1/2$.

Normal form transformation effectively handles the equation's nonlinearity.

The approach advances understanding of well-posedness for dispersive PDEs.

Abstract

We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin-Ono equation is unique in $L^\infty_t(H^s(\mathbb{T}))$ for $s>1/2$. The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.