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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 \(H^{k}\) for every integer \(k\geq 2\). We also infer an additional family of matrix-valued conservation laws of which the previous family is just the traces.

1 Introduction

In a recent paper [1], Berntson, Langmann, and Lenells have introduced the following spin generalization of the Benjamin–Ono equation on the line \({\mathbb{R}}\) or on the torus \({\mathbb{T}}\),

$$\begin{aligned} \partial _{t}U+\{U,\partial _{x} U\} +H\partial _{x}^{2}U -i[U,H \partial _{x}U]=0 ,\quad x\in X, \end{aligned}$$

where X denotes \({\mathbb{R}}\) or \({\mathbb{T}}\), the unknown U is valued into \(d\times d\) matrices, and H denotes the scalar Hilbert transform on X; in fact, the authors chose the normalization \(H=i {\mathrm{sign}}(D)\) so that \(H\partial _{x} =-|D|\), where \(|D|\) denotes the Fourier multiplier associated to the symbol \(|k|\). Notice that in front of the commutator term on the right-hand side, we take a different sign from the one used in [1]. However, passing to the other sign by applying the complex conjugation is easy. Consequently, the above equation reads

$$\begin{aligned} \partial _{t}U=\partial _{x} \bigl( \vert D \vert U-U^{2}\bigr)-i\bigl[U, \vert D \vert U\bigr] . \end{aligned}$$

The purpose of this note is to prove that equation (1) enjoys a Lax pair structure and to infer the first consequences on the corresponding dynamics.

2 The Lax pair structure

Let us first introduce some more notation. Given operators \(A,B\), we denote

$$\begin{aligned} \{ A,B\} :=AB+BA ,\qquad |A,B]:=AB-BA \end{aligned}$$

and \(A^{*}\) denote the adjoint of A. We consider the Hilbert space \(\mathscr{H}:=L^{2}_{+}(X,{\mathbb{C}}^{d\times d})\) made of \(L^{2}\) functions on X with Fourier transforms supported in nonnegative modes, and valued into \(d\times d\) matrices, endowed with the inner product \(\langle A\vert B\rangle =\int _{X} {\mathrm{tr}}(AB^{*}) \,dx \). We denote by \(\Pi _{\geq 0}\) the orthogonal projector from \(L^{2}(X,{\mathbb{C}}^{d\times d})\) onto \(\mathscr{H}\). According to the study of the integrability of the scalar Benjamin–Ono equation [2], given \(U\in L^{2}(X,{\mathbb{C}}^{d\times d})\) valued into \({\mathbb{C}}^{d\times d}\), we define on \(\mathscr{H}\) the unbounded operator

$$\begin{aligned} L_{U}:=D-T_{U} ,\qquad D:=\frac {1}{i} \partial _{x} , \end{aligned}$$

where \({\mathrm{dom}}(L_{U}):=\{ F\in \mathscr{H}: DF\in \mathscr{H}\}\), and \(T_{U}\) is the Toeplitz operator of symbol U defined by \(T_{U}(F):=\Pi _{\geq 0}(UF)\). It is easy to check that \(L_{U}\) is self-adjoint 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,

$$\begin{aligned} B_{U}:=i\bigl(T_{|D|U}-T_{U}^{2}\bigr) , \end{aligned}$$

which is anti-self-adjoint 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 I valued into \(H^{2}(X,{\mathbb{C}}^{d\times d})\) such that \(\partial _{t}U\) is continuous valued into \(L^{2}(X,{\mathbb{C}}^{d\times d})\). Then U is a solution of (1) on I if and only if

$$\begin{aligned} \partial _{t}L_{U}=[B_{U},L_{U}] . \end{aligned}$$


Obviously, \(\partial _{t}L_{U}=-T_{\partial _{t}U}\). Since \(T_{G}=0\) implies classically \(G=0\), the claim is equivalent to the identity

$$\begin{aligned} -T_{\partial _{x} (|D|U-U^{2})-i[U,|D|U]}=[B_{U},L_{U}] . \end{aligned}$$

We have

$$\begin{aligned} -T_{\partial _{x} (|D|U-U^{2})-i[U,|D|U]} &=[iT_{|D|U},D]+T_{U \partial _{x}U+\partial _{x}U U}+iT_{[U,|D|U]} \\ &=[B_{U},D]+T_{U\partial _{x}U+\partial _{x}U U}-T_{U}T_{\partial _{x}U}-T_{ \partial _{x}U}T_{U}+iT_{[U,|D|U]} \\ &=[B_{U},L_{U}]+T_{\{ U,\partial _{x}U\}} -\{ T_{U},T_{\partial _{x}U} \}+iT_{[U,|D|U]}-i[T_{U},T_{|D|U}] \end{aligned}$$

So, we have to check that

$$\begin{aligned} T_{\{ U,\partial _{x}U\}} -\{ T_{U},T_{\partial _{x}U} \}+iT_{[U,|D|U]}-i[T_{U},T_{|D|U}] =0 . \end{aligned}$$

We need the following lemma, where we denote \(\Pi _{<0}:=Id -\Pi _{\geq 0} \).

Lemma 1

Let \(A, B\in L^{\infty}(X,{\mathbb{C}}^{d\times d})\). Then, for every \(F\in \mathscr{H}\),

$$\begin{aligned} (T_{AB}-T_{A}T_{B})F=\Pi _{\geq 0}\bigl( \Pi _{\geq 0}(A) \Pi _{< 0}\bigl(\Pi _{< 0}(B)F\bigr) \bigr) . \end{aligned}$$

Let us prove Lemma 1. Write

$$\begin{aligned} T_{AB}F=\Pi _{\geq 0}(ABF)=\Pi _{\geq 0}\bigl(A\Pi _{\geq 0}(BF)\bigr)+\Pi _{ \geq 0}\bigl(A\Pi _{< 0}(BF) \bigr)=T_{A}T_{B}F+\Pi _{\geq 0}\bigl(A\Pi _{< 0}(BF)\bigr) \end{aligned}$$

so that observing that the ranges of \(\Pi _{\geq 0}\) and of \(\Pi _{<0}\) are stable through the multiplication,

$$\begin{aligned} (T_{AB}-T_{A}T_{B})F=\Pi _{\geq 0}\bigl(A \Pi _{< 0}(BF)\bigr)=\Pi _{\geq 0}\bigl(\Pi _{ \geq 0}(A)\Pi _{< 0}\bigl(\Pi _{< 0}(B)F\bigr)\bigr) . \end{aligned}$$

This completes the proof of Lemma 1. Let us apply Lemma 1 to \(A=U\), \(B=|D|U\). We get

$$\begin{aligned} i(T_{U \vert D \vert U}-T_{U}T_{ \vert D \vert U})F&=\Pi _{\geq 0} \bigl(\Pi _{\geq 0}(U) \Pi _{< 0}\bigl( \Pi _{< 0}\bigl(i \vert D \vert U\bigr)F\bigr)\bigr) \\ &=-\Pi _{\geq 0}\bigl(\Pi _{\geq 0}(U) \Pi _{< 0}\bigl(\Pi _{< 0}(\partial _{x}U)F\bigr)\bigr) , \end{aligned}$$

and similarly

$$\begin{aligned} i(T_{ \vert D \vert U U}-T_{ \vert D \vert U}T_{U})F&=\Pi _{\geq 0} \bigl(\Pi _{\geq 0}\bigl(i \vert D \vert U\bigr) \Pi _{< 0} \bigl(\Pi _{< 0}(U)F\bigr)\bigr) \\ &=\Pi _{\geq 0}\bigl(\Pi _{\geq 0}(\partial _{x}U) \Pi _{< 0}\bigl(\Pi _{< 0}(U)F\bigr)\bigr) \end{aligned}$$

so that

$$\begin{aligned} \bigl(iT_{[U,|D|U]}-i[T_{U},T_{|D|U}]\bigr)F={}&{-}\Pi _{\geq 0}\bigl(\Pi _{\geq 0}(U) \Pi _{< 0}\bigl(\Pi _{< 0}(\partial _{x}U)F\bigr)\bigr) \\ &{}-\Pi _{\geq 0}\bigl(\Pi _{\geq 0}(\partial _{x}U) \Pi _{< 0}\bigl(\Pi _{< 0}(U)F\bigr)\bigr) \\ ={}&{-}T_{\{ U,\partial _{x}U\}} (F)+\{ T_{U}, T_{\partial _{x}U}\} (F) , \end{aligned}$$

using again Lemma 1. Hence, we have proved identity (2).  □

3 Conservation laws and global wellposedness

The following is an application of Theorem 1.

Corollary 1

Assume that \(U_{0}\) belongs to the Sobolev space \(H^{2}(X,{\mathbb{C}}^{d\times d})\) and is valued into Hermitian matrices. Then equation (1) has a unique solution U, depending continuously on \(t\in {\mathbb{R}}\), 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,

$$\begin{aligned} \mathscr{E}_{k}(U)=\bigl\langle L_{U}^{k}(\Pi _{\geq 0} U)\vert \Pi _{\geq 0}U \bigr\rangle ,\quad k=0,1,2\dots. \end{aligned}$$

In particular, the norm of \(U(t)\) in the Sobolev space \(H^{2}(X)\) is uniformly bounded for \(t\in {\mathbb{R}}\).


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 \(U_{+}:=\Pi _{\geq 0}U , U_{-}:=\Pi _{<0}U \). Applying \(\Pi _{\geq 0}\) to both sides of (1), we get

$$\begin{aligned} \partial _{t}U_{+}=-i\partial _{x}^{2}U_{+}-2T_{U} \partial _{x}U_{+}-2T_{ \partial _{x}U_{-}}U_{+}=iL_{U}^{2}(U_{+})+B_{U}(U_{+}) . \end{aligned}$$

Therefore, from Theorem 1,

$$\begin{aligned} \frac{d}{dt}\bigl\langle L_{u}^{k} (U_{+}) \vert U_{+}\bigr\rangle ={}&\bigl\langle \bigl[B_{U},L_{U}^{k} \bigr]U_{+} \bigl\vert U_{+}\bigr\rangle +\bigl\langle L_{U}^{k}\bigl(iL_{U}^{2}(U_{+})+B_{U}(U_{+}) \bigr) \bigr\vert U_{+}\bigr\rangle \\ &{} +\bigl\langle L_{U}^{k}(U_{+})\vert iL_{U}^{2}(U_{+})+B_{U}(U_{+}) \bigr\rangle \\ ={}& 0 , \end{aligned}$$

since \(B_{U}\) and \(iL_{U}^{2}\) are anti-self-ajoint.

Now observe that \(\mathscr{E}_{0}(U)=\| U_{+}\|_{L^{2}}^{2}\). Since U is Hermitian, we have

$$\begin{aligned} U= \textstyle\begin{cases} U_{+}+ U_{+}^{*} &{\text{if }} X={\mathbb{R}} , \\ U_{+}+U_{+}^{*} -\langle U_{+}\rangle &{\text{if }} X= {\mathbb{T}} , \end{cases}\displaystyle \end{aligned}$$

where \(\langle F\rangle \) denotes the mean value of a function F on \({\mathbb{T}}\). We infer that \(\mathscr{E}_{0}(U)\) controls the \(L^{2}\) norm of U. Let us come to \(\mathscr{E}_{1}(U)\). In view of the Gagliardo–Nirenberg inequality,

$$\begin{aligned} \mathscr{E}_{1}(U)&=\bigl\langle DU_{+} \bigl\vert U_{+}\bigr\rangle -\bigl\langle T_{U}(U_{+}) \bigr\vert U_{+}\bigr\rangle \geq \langle DU_{+}\vert U_{+}\rangle -O\bigl( \Vert U_{+} \Vert _{L^{3}}^{3}\bigr) \\ &\geq \langle DU_{+}\vert U_{+}\rangle - O\bigl(\langle DU_{+}\vert U_{+} \rangle ^{1/2} \Vert U_{+} \Vert _{L^{2}}^{2}\bigr)-O\bigl( \Vert U_{+} \Vert _{L^{2}}^{3}\bigr) . \end{aligned}$$

Consequently, \(\mathscr{E}_{0}(U)\) and \(\mathscr{E}_{1}(U)\) control \(\| U_{+}\|_{L^{2}}^{2}+\langle DU_{+}\vert U_{+}\rangle \), 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 \(\mathscr{E}_{0}(U)\) and \(\mathscr{E}_{1}(U)\).

Since \(\mathscr{E}_{2}(U)\) is the square of \(L^{2}\) norm of \(L_{U}(U_{+})\) and 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 \(\mathscr{E}_{0}(U)\), \(\mathscr{E}_{1}(U)\), and \(\mathscr{E}_{2}(U)\) control the \(L^{2}\) norms of U and of \(\partial _{x}U\), namely the Sobolev \(H^{1}\) norm of U.

Finally, \(\mathscr{E}_{4}(U)\) is the square if the \(L^{2}\) norm of \(L_{U}^{2}(U_{+})\). Since \(L_{U}(U_{+})\) is already controlled in \(L^{2}\) and U is controlled in \(L^{\infty }\) by the Sobolev inclusion \(H^{1}\subset L^{\infty }\), we infer that the \(H^{1}\) norm of \(L_{U}(U_{+})\) is controlled. But \(H^{1}\) is an algebra, so the \(H^{1}\) norm of \(T_{U}(U_{+})\) is also controlled. Finally, we infer that \(\{ \mathscr{E}_{n}(U), n\leq 4\}\) control the \(H^{1}\) norms of \(U_{+}\) and \(\partial _{x}U_{+}\), namely the \(H^{2}\) norm of \(U_{+}\), and finally of U. □


  1. (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 \(\{ \mathscr{E}_{n} (U), 0\leq n\leq 2k\}\).

  2. (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 \(X={\mathbb{T}}\)—is left open. Since Corollary 1 implies that the \(L^{\infty }\) 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 conservation laws. The conservation laws \(\mathscr{E}_{k}\) can be explicitly computed in terms of U. For simplicity, we focus on \(\mathscr{E}_{0}\) and \(\mathscr{E}_{1}\). In case \(X={\mathbb{R}}\), we have exactly

$$\begin{aligned} \mathscr{E}_{0}(U)=\frac {1}{2} \int _{{\mathbb{R}}}{\mathrm{tr}}\bigl(U^{2}\bigr) \,dx , \end{aligned}$$


$$\begin{aligned} \mathscr{E}_{1}(U)&=\bigl\langle DU_{+} \bigl\vert U_{+}\bigr\rangle -\bigl\langle T_{U}(U_{+}) \bigr\vert U_{+}\bigr\rangle \\ &= \int _{{\mathbb{R}}}{\mathrm{tr}} \biggl( \frac {1}{2} U \vert D \vert U - \frac {1}{3} U^{3} \biggr) \,dx , \end{aligned}$$

so we recover the Hamiltonian function derived in [1].

In case \(X={\mathbb{T}}\), 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 \(V\in {\mathbb{C}}^{d\times d}\) is a special element of \(\mathscr{H}\), and we observe that \(B_{U}(V)=-iL_{U}^{2}(V)\). Arguing exactly as in the proof of Corollary 1, we infer that, for every integer \(\ell \geq 1\), for every pair of constant matrices \(V,W\), the quantity \(\langle L_{U}^{\ell }(V)\vert W\rangle \) is a conservation law. Since \(V,W\) are arbitrary, this means that, if 1 denotes the identity matrix, all the matrix-valued functionals

$$\begin{aligned} \mathscr{M}_{\ell -2}(U):= \int _{{\mathbb{T}}} L_{U}^{\ell} ({\mathbf{1}}) \,dx \end{aligned}$$

for \(\ell \geq 1\) are conservation laws. If the measure of \({\mathbb{T}}\) is normalised to 1, we have for instance

$$\begin{aligned} &\mathscr{M}_{-1}(U)=-\langle U_{+}\rangle =-\langle U \rangle , \\ &\mathscr{M}_{0}(U)=\frac {1}{2}\bigl\langle U^{2}-iUHU\bigr\rangle + \frac {1}{2}\langle U\rangle ^{2} . \end{aligned}$$

Then one can check that

$$\begin{aligned} &\mathscr{E}_{0}(U)= \frac {1}{2}{\mathrm{tr}}\bigl(\bigl\langle U^{2}\bigr\rangle \bigr)+ \frac {1}{2} {\mathrm{tr}}\bigl(\langle U\rangle ^{2}\bigr) , \\ &\mathscr{E}_{1}(U)= {\mathrm{tr }} \biggl\langle \frac {1}{2} U \vert D \vert U - \frac {1}{3} U^{3} \biggr\rangle - \frac {5}{3} {\mathrm{tr}} \bigl[\langle U \rangle ^{3}\bigr] -{ \mathrm{tr}}\bigl[\mathscr{M}_{0}(U) \langle U\rangle \bigr] . \end{aligned}$$

Observe again that the first term on the right-hand side of the expression of \(\mathscr{E}_{1}(U)\) is the opposite of the Hamiltonian function in [1].

In the case \(X={\mathbb{R}}\), all the matrix valued expressions \(\mathscr{M}_{k}(U)\) make sense if \(k\geq 0\) and are again conservation laws. For instance,

$$\begin{aligned} \mathscr{M}_{0}(U)=\frac {1}{2} \int _{{\mathbb{R}}}\bigl( U^{2}-iUHU\bigr) \,dx . \end{aligned}$$

Finally, notice that in both cases \(X={\mathbb{T}}\) and \(X={\mathbb{T}}\), we have

$$\begin{aligned} \mathscr{E}_{k}(U)={\mathrm{tr}}\mathscr{M}_{k}(U) \end{aligned}$$

for every \(k\geq 0\).

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  1. Berntson, B., Langmann, E., Lenells, J.: Spin generalizations of the Benjamin–Ono equation. arXiv:2201.07269v1

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The author is grateful to Edwin Langmann for drawing his attention to equation (1) and for stimulating discussions.


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Gérard, P. The Lax pair structure for the spin Benjamin–Ono equation. Adv Cont Discr Mod 2023, 21 (2023).

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