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 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

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

(1)

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

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

Proof

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

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

(2)

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 $t\in {\mathbb{R}}$.

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 $U_{+}:=\Pi _{\geq 0}U , U_{-}:=\Pi _{<0}U$. 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. □

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

• (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 $X={\mathbb{T}}$—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

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

and

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

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 $k\geq 0$.