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

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.