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On the hamiltonian formulation of an octonionic integrable extension for the Korteweg-de Vries equation

We present in this work the hamiltonian formulation of an octonionic extension for the Korteweg-de Vries equation. The formulation takes into account the non commmutativity and non associativity of the implicit algebra which defines the equation. We also analize the Poisson structure of the hamiltonian formulation. We propose a parametric master Lagrangian which contains the two hamiltonian structures of the integrable octonionic equation.


Introduction
Several extensions of the Korteweg-de Vries (KdV) equation have been proposed since its introduction in [1,2,3,4,5]. Also many interesting developments on integrable systems have been obtained since then. Besides, an all time challenge in theoretical physics has been to understand the role of the octonion algebra, the unique nonassociative and noncommutative real normalized division algebra in the modelling of the fundamental interactions in nature. In particular, it is well known the relation of the octonions with the maximal supersymmetric Yang-Mills theory in ten dimensions [6] and the Supermembrane theory in eleven dimensions in the context of theories searching for the unification of all fundamental interactions.
Recently, an extension of the KdV equation with fields valued on a general Cayley-Dickson algebra, in particular the octonion algebra, was proposed [7]. The Bäcklund transformation, Gardner parametric system, Lax pair and an infinite sequence of conserved quantities were obtained. In this paper we analyse the Hamiltonian structure of that extension for the fields valued on the octonion algebra.

Korteweg-de Vries equation valued on the algebra of octonions
We denote by u = u(x, t) a function valued on the octonionic algebra O.
If we denote by e i (i=1,. . . ,7), the imaginary basis of the octonions, u can be expressed by where b(x, t) and The Korteweg-de Vries (KdV) equation formulated on the algebra of octonions, with a one non trivial term can be given by when − → B = − → 0 it reduces to the scalar KdV equation. The field v is an octonionic constant which can be interpreted as a external field.
In [7] equation (2) was formulated in the context of a Cayley-Dickson algebra as an integrable system (in the sense of having an infinite sequence of polynomial conserved quantities).
In terms of b and − → B the equation can be re-expressed as Equation (2) is invariant under Galileo transformations. In fact, if where c is a real constant, then ∂ ∂x = ∂ ∂ x and ∂ ∂t = c ∂ ∂ x + ∂ ∂ t . Additionally, equation (2) is invariant under the subgroup of the group of automorphisms of the octonions G 2 which leaves v invariant.
If under an automorphism we have u → φ(u) and consequently, The subgroup of G 2 with that property is SU (3). Hence, the KdV extension (2) is invariant under the group SU (3).

The master Lagrangian for the octonionic KdV equation
From now on we do not include the bracket term [v, u], that is, we consider v = 0. We will consider the complete extension elsewhere.
We use now the Hemholtz procedure to obtain a Lagrangian density for the generalized Gardner equation defined by (5) and (2) are related through in the case v = 0. Thanks to (6), any solution of (5) gives a solution of (2).
From it and following the construction in [8] we obtain the two Lagrangians associated to the octonionic KdV equation. The master Lagrangian formulated in terms of the Gardner prepotential s(x, t) , where the Lagrangian density is given by Independent variations with respect to s yields the generalized Gardner equation [7], as expected since it is guaranteed by the Helmholtz procedure.
The Lagrangian density L (s) is invariant under the action of the exceptional Lie group G 2 . In fact, consider the infinitesimal G 2 transformation where λ ij are the real infinitesimal parameters and D e i ,e j (s) are the generators of the associated derivation algebra, the Lie algebra of the Lie group G 2 . Its definition and properties are discussed, for example in [7].
We get we also have (δs) x = λ ij D e i ,e j (s x ) = δs x . Using now the Leibnitz rule for D e i ,e j we obtain but D e i ,e j ( ) is pure imaginary for any argument, hence δL (s) = 0. If we take the limit → 0, we obtain a fisrt Lagrangian for the octonionic KdV equation, Independent variations with respect to w yields, using u = w x , the octonionic KdV equation (2). If we take the following redefinition We get in this limit the generalized Miura Lagrangian The Miura equation is then obtained by taking variations with respect toŝ, we get The Lagrangian formulation of the octonionic KdV equation may be used as the starting step to obtain the hamiltonian structures of the octonionic system.
We may now construct, using de Dirac approach [9] for constrained systems, the two hamiltonian structures associated to the lagrangians L and L M . This approach was used in [10,11] to obtain the, previously known, first and second hamiltonian structures of KdV equation. Applications of Dirac's procedure to obtain the corresponding Poisson structures in another extensions of KdV type can be seen in [8,12].
We first consider the Lagrangian L and define the conjugate momenta associated to w 0 and w i (i = 1, . . . , 7) defined by w 0 = Re(w) w i e i = Im(w), w i , i = 1, . . . , 7; summation in the repeated index is understood.
We denote p and p i the conjugate momenta of w 0 and w i respectively. We have, from the definition of the momenta, They define constraints on the phase space These are second class constraints. In fact, In order to obtain the Poisson structure on the constrained phase space we use the Dirac theory of constraints. The Dirac bracket between two functionals F and G is where {} x denotes integration on x from −∞ to +∞. The indices m, n range from 0 to 7 and C mn are the components of the inverse of the matrix whose components are the poisson brackets of the second class constraints (7). The Hamiltonian is obtained from the Lagrangian L through a Legendre transformation, where u 0 is the real part of the octonionic field u and u i the components of its imaginary part. Summation on repeated indices is understood. The Poisson structure on the constrained phase space is obtained from the Dirac brackets. We obtain