A new integrable structure associated to the Camassa-Holm peakons

We provide a closed Poisson algebra involving the Ragnisco--Bruschi generalization of peakon dynamics in the Camassa--Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a one-parameter integrable extension of their structure. It leads to a new $N$-body peakon solution to the Camassa--Holm shallow-water equation depending on two parameters. We present two explicit constructions of a (non-dynamical) $r$-matrix formulation for this new Poisson algebra. The first one relies on a tensorization of the $N$-dimensional auxiliary space by a 4-dimensional space. We identify a family of Poisson commuting quantities in this framework, including the original ones. This leads us to constructing a second formulation identified as a spectral parameter representation.


Introduction
Many non-linear two-dimensional (x, t) integrable fluid equations exhibit so-called peakon solutions which take the generic form u(x, t) = N i=1 p i (t) e −|x−q i (t)| . (1) Their dynamics for (p i , q i ) is deduced from a reduction of the 1+1 fluid equations for u(x, t) and may have integrability properties.The best known example of such integrability feature for peakon equations is given by the Camassa-Holm shallow-water equation [5,6] u t − u x x t + 3uu x = 2u x u x x + uu x x x . ( Integrability of the peakons themselves was studied in particular in [7].A one-parameter extension was proposed by Ragnisco and Bruschi [8], who proved integrability from an implicit construction of a dynamical r-matrix.An explicit construction of this dynamical r-matrix is still lacking.In this paper, we propose an alternative construction of the Poisson structure relevant for integrability.It relies on a non-dynamical r-matrix formulation and uses three dynamical Lax-type matrices, instead of the two matrices introduced by Ragnisco and Bruschi.A direct consequence is that the new integrable peakon model depends on two coupling constants.It provides a more general peakon dynamics which in turns yields a new peakon-type solution of the Camassa-Holm equation (2).We present two representations of the non-dynamical r-matrix formulation.The first one requires the use of a larger auxiliary space obtained by tensoring the initial N -dimensional space by a 4-dimensional space.One advantage of this representation is that the Yang-Baxter equation for the r-matrix structure takes a remarkably simple and compact form.The second representation involves the introduction of a spectral parameter in the N -dimensional auxiliary space.Both formulations provide an algebraic framework to construct the same hierarchy of conserved quantities.
The plan of the paper runs as follows.Section 2 describes the complete Poisson algebra of the three relevant matrix generators involved in our formulation of the integrability properties.It allows the construction of a family of Poisson commuting quantities, including a two-parameter generalization of the peakon Hamiltonian.From the N -body dynamics triggered by this new Hamiltonian we deduce a new peakon solution of the Camassa-Holm equation.
Section 3 is devoted to the construction of the r-matrix in the extended auxiliary space picture.The explicit computation of the classical Yang-Baxter equation is performed.It is in fact a modified classical Yang-Baxter equation, where the r.h.s. is built from the three-fold tensored Casimir operator of the algebra.
Finally, Section 4 displays the construction of Poisson commuting quantities.It is first done in the framework of the extended auxiliary space.The existence of families of Poisson commuting quantities naturally induces the existence of an alternative, spectral parameter presentation.

Poisson structure 2.1 Description of the original model and its first generalisation
The n-peakon solutions (1) of the Camassa-Holm equation yield a dynamical system for p i , q i : qi where s i j = sgn(q i − q j ).This discrete dynamical system is described by a Hamiltonian with the canonical Poisson structure: The dynamics is encoded in the Lax formulation [4,5] with where E i j is the N × N elementary matrix with 1 at position (i, j) and 0 elsewhere.The Hamiltonian (4) is recast as H C H =1 2 TrL 2 .A generalisation of this original integrable model was proposed by Ragnisco-Bruschi [8], with a Lax matrix 1 L(ρ) = T + ρS, ρ ∈ , where The proof of integrability relies on the construction of a dynamical r-matrix [8].The Hamiltonian takes the form One recovers the original Hamiltonian H C H for the values ρ = −1, ν = 1 or ρ = 1, ν = −1.

General non dynamical Poisson structure
The description of the full algebraic structure associated to the Poisson brackets (6) and the Lax matrix L(ρ) requires the introduction of a third matrix which allows to close the Poisson structure of (T, S).It reads with In ( 13)-( 19), we have used the auxiliary space description: for any N × N matrix M , we define Remark that, following [2], Γ 12 can be identified with a classical r-matrix for the classical open Toda chain.

Generalized peakons
From the above Poisson structure, it is natural to introduce the most general Lax matrix hereafter denoted L. The Poisson structure of this Lax matrix reads We ask the traces τ n = Tr( Ln ) to be Poisson commuting for all values of n ∈ + .A direct calculation leads to We first notice that for any matrix M we have where we have used that for any matrices U, V , to get the second line), the property Tr 2 Π = N (third line) and the cyclicity of the trace (fourth line).
Similarly, we have Tr 12 ( L1 ) n−1 ( L2 ) m−1 Π, M 2 = 0, so that when computing {τ n , τ m }, the terms corresponding to Π in ( 22) can be dropped.Thanks to this property, we get where r 12 = ρ Γ 12 + λ Π t .Now, for any matrix M we have Π t , M 2 = Π t , M t 1 and since A is an antisymmetric matrix, we get (26) Furthermore, starting from the relations valid for any matrix R 12 , we get when Finally, using that we deduce that Π t , L1 − L2 = 2λ Π t , A 1 , so that from (28) we get Then, (26) rewrites as Thus, the model associated to L defines an integrable double deformation of the original peakon model.
Remark that a generic value for ν can be obtained from the cases ν = ±1, see section 2.4 below.Hence the above conditions correspond to a condition on ρ and λ, rather than two conditions on ρ, λ and ν.

N-body solutions of the fluid equation
We establish here a general result on the consistency conditions for peakon-type N -body solutions to (a deformation of) the Camassa-Holm equation, including the Hamiltonian evolution of the N -body variables.We first consider the case ν 2 = 1, and then show how a generic value for ν can be obtained from the cases ν = ±1.

A deformed version of the Camassa-Holm equation
We first restrict ourself to the case The Hamiltonian H new (ρ, λ) given in (32), can be rewritten as with Note that the function F in (37) obey the following differential equation with the initial value conditions where we have used the property ν 2 = 1.
The Hamiltonian H new (ρ, λ) describes a time evolution for p i , q i , given by: where s i j = sgn(q i − q j ) as above, and To recover the time evolutions (39) from a fluid equation, we define u(x, t) as which is a direct generalisation of (1).Plugging the form (41) into the l.h.s. of the differential relation (38) and using (33), one finds with We have introduced the notation From the differential equations (38) and the equations of motion (39), we get To obtain this relation, we have used the property (deduced from the equations of motion (39)) that p defined in (33) is a free constant parameter of the model, i.e. ṗ = 0. Now remarking that N i=1 sgn(x − q i )p i F ′ i = u x , we find a modification of the Camassa-Holm shallow-water equation For µ = 0 we recover the undeformed Camassa-Holm shallow-water equation.For µ ̸ = 0, we find the deformed fluid equation ( 46).It exhibits peakon-type solutions (41) with the integrable dynamics (32).
Then, the only non-vanishing o i jk 00 ′ 0 ′′ are given by where we omitted the subscript 00 ′ 0 ′′ to lighten the writing.
To obtain (63), we have used the classical Yang-Baxter equation for Γ [1]: Remark that the formulas (63) imply the following formula for O I,II,III : e j j ⊗ e j j ⊗ e j j , Z = e 12 + e 21 + e 34 + e 43 . (65) We have defined t 1 = t 0 , t 2 = t 0 ′ , t 3 = t 0 ′′ .The notation Z s stands for the matrix Z acting in the auxiliary space s.

Conserved quantities from the operator
Now that we have established a Lax presentation of the Poisson brackets, we can consider the traces where K = i k i e ii ⊗ N is a diagonal 4N × 4N matrix, acting as identity in the spaces 0 and 0 ′ .We have denoted "Tr" the trace in the N -dimensional space and "tr" the trace in the 4dimensional space.Since all matrices commute in the 4-dimensional space, it is easy to show that {t As a consequence we have a commuting family of operators generated by Since A is antisymmetric while T and S are symmetric, we get using TrM = Tr(M T ): To get more insight on the two remaining generators, we consider the first chamber Then, looking at the form of A and S, one sees that A + S is a triangular matrix with zeros on the diagonal, so that Tr Similarly, a simple recursion shows that We then get Tr where p is given in (33).
Since we obtain the other chambers through permutation of rows and columns, the properties (71) and ( 73) are valid everywhere.In conclusion, the quantities t K n provide only one conserved quantity p and its corresponding polynomial algebra.

Adding more conserved quantities
It is easy to show that since 2T = ℓ 1 − ℓ 2 , 2S = ℓ 3 − ℓ 4 , and 2A = ℓ 1 + ℓ 2 = ℓ 3 + ℓ 4 , we have which is valid for any δ ∈ .Then, one recovers the conserved quantities τ n as Note that to prove this property, we need to consider trD n instead of n , which explains why the "natural" approach using the elements t K n does not lead to the full set of conserved quantities.
Remark that when D is replaced by a general diagonal matrix K, the set is still commutative: {τ K n , τ K m } = 0. We now consider two diagonal matrices K and K ′ and look for the vanishing of the Poisson brackets {τ K ′ n , τ K m }.Using a formal computation software, we are led to propose a family of matrices 8ρ(x + 1) , 0 , depending on a parameter x, such that: Note that there are other choices of K matrices leading to the same hierarchy of conserved quantities τ n (x).The analytical proof of this property is given at the end of section 4.3.We first provide a spectral parameter formulation of the Poisson brackets.

A spectral parameter representation of the Poisson structure
For fixed λ and ρ, we introduce a family extending the L operator ( 21) Indeed one can check from (77) that we have trK(x) = T +ā(x) S+a(x)A ≡ L(x) , with ) .
(80) The relation in (79) follows from the explicit form of β 1 (x) and β 2 (x).Some specific values for x provide interesting sub-cases: L(x 0 ) corresponds to the original L matrix, while L(x ± ) leads to the Ragnisco-Bruschi Lax matrix L(ρ).The last case corresponds to a deformation of the original Peakon Lax matrix in a direction orthogonal to the one chosen by Ragnisco-Bruschi.We now establish the Poisson bracket structure between any two matrices of this family.From the Poisson brackets (13)-( 19), we obtain .Then one identifies consistently Finally, using once more the relations Π, M 2 = − Π, M 1 and Π t , M t 2 = Π t , M 1 (for any matrix M ), we can rewrite the above relation in a r-matrix form: Since τ n a(x), ā(x) ≡ τ n (x), this ends the proof of property (78).