The Lax pair structure for the spin Benjamin–Ono equation

We prove that the recently introduced spin Benjamin–Ono equation admits a Lax pair and deduce a family of conservation laws that allow proving global wellposedness in all Sobolev spaces Hk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{k}$\end{document} for every integer k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\geq 2$\end{document}. We also infer an additional family of matrix-valued conservation laws of which the previous family is just the traces.


Introduction
In a recent paper [1], Berntson, Langmann and Lenells have introduced the following spin generalization of the Benjamin-Ono equation on the line R or on the torus T, x U − i[U, H∂ x U] = 0 , x ∈ X, where X denotes R or T, the unknown U is valued into d × d matrices, and H denotes the scalar Hilbert transform on X ; in fact, the authors chose the normalization H = i sign(D), so that H∂ x = −|D|, where |D| denotes the Fourier multiplier associated to the symbol |k|.Notice that, in front of the commutator term in the right hand side, we take a different sign from the one used in [1].However it is easy to pass to the other sign by applying the complex conjugation.Consequently, the above equation reads (1) ∂ The purpose of this note is to prove that equation (1) enjoys a Lax pair structure, and to infer first consequences on the corresponding dynamics.

The Lax pair structure
Let us first introduce some more notation.and A * denote the adjoint of A. We consider the Hilbert space H := L 2 + (X, C d×d ) made of L 2 functions on X with Fourier transforms supported in nonnegative modes, and valued into d × d matrices, endowed with the inner product A|B = X tr(AB * ) dx .We denote by Π ≥0 the orthogonal projector from L 2 (X, C d×d ) onto H .
According to the study of the integrability of the scalar Benjamin-Ono equation [2], given U ∈ L 2 (X, C d×d ) valued into C d×d , we define on H the unbounded operator where dom(L U ) := {F ∈ H : DF ∈ H }, and T U is the Toeplitz operator of symbol U defined by T U (F ) := Π ≥0 (UF ) .
It is easy to check that L U is selfadjoint if U is valued in Hermitian matrices.However we do not need the latter property for establishing the Lax pair structure.If U is smooth enough (say belonging to the Sobolev space H 2 ), we define the following bounded operator, , which is antiselfadjoint if U is valued in Hermitian matrices.Our main result is the following.Theorem 1.Let I be a time interval, and U be a continuous function on Then U is a solution of (1) on I if and only if We need the following lemma, where we denote Π <0 := Id − Π ≥0 .
Let us prove Lemma 1. Write so that, observing that the ranges of Π ≥0 and of Π <0 are stable through the multiplication, This completes the proof of Lemma 1.

Let us apply Lemma 1 to
and similarly using again Lemma 1. Hence we have proved identity (2).

Conservation laws and global wellposedness
The following is an application of Theorem 1.
Corollary 1. Assume U 0 belongs to the Sobolev space H 2 (X, C d×d ), and is valued into Hermitian matrices.Then equation (1) has a unique solution U, depending continuously of t ∈ R, valued into Hermitian matrices of the Sobolev space H 2 (X), and such that U(0) = U 0 .Furthermore, the following quantities are conservation laws, . .In particular, the norm of U(t) in the Sobolev space H 2 (X) is uniformly bounded for t ∈ R.
Proof.The local wellposedness in the Sobolev space H 2 follows from an easy adaptation of Kato's iterative scheme -see e.g.Kato [3] for hyperbolic systems.Global wellposedness will follow if we show that conservation laws control the H 2 norm.Set U + := Π ≥0 U , U − := Π <0 U .Applying Π ≥0 to both sides of (1), we get Therefore, from Theorem 1, Since U is Hermitian, we have , where F denotes the mean value of a function F on T. We infer that E 0 (U) controls the L 2 norm of U. Let us come to E 1 (U).In view of the Gagliardo-Nirenberg inequality, which is the square of the H 1/2 norm of U + , since U + only has nonnegative Fourier modes.Therefore the H 1/2 norm of U is controlled by E 0 (U) and E 1 (U).Since E 2 (U) is the square of L 2 norm of L U (U + ), and since the L 2 norm of T U (U + ) is controlled by the H 1/2 norm of U by the Sobolev estimate, we infer that E 0 (U), E 1 (U) and E 2 (U) control the L 2 norms of U and of ∂ x U, namely the Sobolev H 1 norm of U.
But H 1 is an algebra, so the H 1 norm of T U (U + ) is also controlled.Finally, we infer that {E n (U), n ≤ 4} control the H 1 norms of U + and ∂ x U + , namely the H 2 norm of U + , and finally of U.

Remarks.
(1) If the initial datum U belongs to the Sobolev space H k for an integer k > 2, a similar argument shows that the H k norm of U is controlled by the collection {E n (U), 0 ≤ n ≤ 2k}.(2) In [1], the evolution of multi-solitons for (1) is derived through a pole ansatz, and the question of keeping the poles away from the real line -or from the unit circle in the case X = T-is left open.Since Corollary 1 implies that the L ∞ norm of the solution stays bounded as t varies, this implies a positive answer to this question, as far as the poles do not collide.In fact, we strongly suspect that such a collision does not affect the structure of the pole ansatz, because it is likely that multisolitons have a characterization in terms of the spectrum of L U , as it has in the scalar case [2].Let us say a few more about the conservation laws.The conservation laws E k can be explicitly computed in terms of U.For simplicity, we focus on E 0 and E 1 .In the case X = R, we have exactly so we recover the Hamiltonian function derived in [1].
In the case X = T, the above formulae must be slightly modified due the zero Fourier mode.This leads us to a bigger set of conservation laws.Indeed, every constant matrix V ∈ C d×d is a special element of H , and we observe that B U (V ) = −iL 2 U (V ) .Arguing exactly as in the proof of Corollary 1, we infer that, for every integer ℓ ≥ 1, for every pair of constant matrices V, W , the quantity L ℓ U (V )|W is a conservation law.Since V, W are arbitrary, this means that, if 1 denotes the identity matrix, all the matrix-valued functionals Then one can check that Observe again that the first term in the right hand side of the expression of E 1 (U) is the opposite of the Hamiltonian function in [1].
In the case X = R, all the matrix valued expressions M k (U) make sense if k ≥ 0 and are again conservation laws.For instance,

M ℓ− 2 (
dx for ℓ ≥ 1, are conservation laws.If the measure of T is normalised to 1, we have for instance