The third order Benjamin-Ono equation on the torus : well-posedness, traveling waves and stability

TL;DR

This paper studies the well-posedness, traveling wave solutions, and their stability for the third order Benjamin-Ono equation on the torus, revealing the regularity thresholds for the flow map and classifying stable traveling waves.

Contribution

It establishes the well-posedness and ill-posedness thresholds for the flow map, classifies traveling wave solutions, and analyzes their orbital stability for the third order Benjamin-Ono equation.

Findings

Flow map extends continuously for s ≥ 0

Flow map does not extend continuously for 0 < s < 1/2

Traveling waves are classified and their stability is studied

Abstract

We consider the third order Benjamin-Ono equation on the torus $\partial_t u= \partial_x \left( -\partial_{xx}u-\frac{3}{2}u H\partial_x u - \frac{3}{2}H(u\partial_x u) + u^3 \right).$ We prove that for any $t\in\mathbb{R}$, the flow map continuously extends to $H^s_{r,0}(\mathbb{T})$ if $s\geq 0$, but does not admit a continuous extension to $H^{-s}_{r,0}(\mathbb{T})$ if $0<s<\frac{1}{2}$. Moreover, we show that the extension is not weakly sequentially continuous in $L^2_{r,0}(\mathbb{T})$. We then classify the traveling wave solutions for the third order Benjamin-Ono equation in $L^2_{r,0}(\mathbb{T})$ and study their orbital stability.