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The Lax pair structure for the spin Benjamin–Ono equation
Advances in Continuous and Discrete Models volume 2023, Article number: 21 (2023)
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 matrixvalued 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 \({\mathbb{T}}\),
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 righthand 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
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
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
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 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 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
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
We have
So, we have to check that
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 \(F\in \mathscr{H}\),
Let us prove Lemma 1. Write
so that observing that the ranges of \(\Pi _{\geq 0}\) and of \(\Pi _{<0}\) are stable through the multiplication,
This completes the proof of Lemma 1. Let us apply Lemma 1 to \(A=U\), \(B=DU\). We get
and similarly
so that
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,
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
Therefore, from Theorem 1,
since \(B_{U}\) and \(iL_{U}^{2}\) are antiselfajoint.
Now observe that \(\mathscr{E}_{0}(U)=\ U_{+}\_{L^{2}}^{2}\). Since U is Hermitian, we have
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,
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 multisolitons 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
and
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 matrixvalued functionals
for \(\ell \geq 1\) are conservation laws. If the measure of \({\mathbb{T}}\) is normalised to 1, we have for instance
Then one can check that
Observe again that the first term on the righthand 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,
Finally, notice that in both cases \(X={\mathbb{T}}\) and \(X={\mathbb{T}}\), we have
for every \(k\geq 0\).
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References
Berntson, B., Langmann, E., Lenells, J.: Spin generalizations of the Benjamin–Ono equation. arXiv:2201.07269v1
Gérard, P., Kappeler, T.: On the integrability of the Benjamin–Ono equation on the torus. Commun. Pure Appl. Math. 74, 1685–1747 (2021)
Kato, T.: Abstract evolution equations, linear and quasilinear, revisited. In: Functional Analysis and Related Topics, 1991, Kyoto. Lecture Notes in Math., vol. 1540, pp. 103–125. Springer, Berlin (1993)
Acknowledgements
The author is grateful to Edwin Langmann for drawing his attention to equation (1) and for stimulating discussions.
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À la mémoire de Jean Ginibre
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Gérard, P. The Lax pair structure for the spin Benjamin–Ono equation. Adv Cont Discr Mod 2023, 21 (2023). https://doi.org/10.1186/s13662023037682
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DOI: https://doi.org/10.1186/s13662023037682