The Lax pair structure for the spin Benjamin--Ono equation

TL;DR

This paper establishes a Lax pair for the spin Benjamin--Ono equation, leading to conservation laws that ensure global well-posedness in all Sobolev spaces for integer orders.

Contribution

It introduces a Lax pair structure for the spin Benjamin--Ono equation and derives conservation laws that guarantee global solutions in Sobolev spaces.

Findings

Existence of a Lax pair for the spin Benjamin--Ono equation

Derivation of conservation laws ensuring global well-posedness

Global well-posedness in all Sobolev spaces $H^k$, $k eq 0$

Abstract

We prove that the recently introduced spin Benjamin--Ono equation admits a Lax pair, and we deduce a family of conservation laws which allow to prove global wellposedness in all Sobolev spaces $H^k$ for every integer $k\geq 2$. We also infer an additional family of matrix valued conservation laws, of which the previous family are just the traces.