Hamiltonian formulation of SL(3) Ur-KdV equation

We give a unified view of the relation between the SL(2) KdV, the mKdV, and the UrKdV equations through the Fréchet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no nonlocal operators. We extend this method to the SL(3) KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bsq equation in a simple form. In particular, we explicitly construct the hamiltonian operator of the Ur-Bsq system which defines the poisson structure of the system, through the Fréchet derivative and its inverse. ∗E-mail address: [email protected] Integrable models[1] in two dimensions in terms of nonlinear differential equations have many interesting properties. In particular, they are bi-hamiltonian systems, which means there are at least two distinct hamiltonian structures[2]. Such a structure provides the integrability to the system through infinite number of conserved quantities in involution. Another interesting feature of the integrable system is the existence of the zero curvature condition, i.e., Lax equation which is a compatibility condition for the pair of auxiliary linear equations. Lax equation is crucical in solving the integrable system by the inverse scattering method. Recently relations between the integrable nonlinear differential equations and two dimensional conformal field theories drew a lot of attention. For example, the poisson brackets of the second hamiltonian stucture of the KdV equation is the classical version of the Virasoro algebra[3]. More generally, it was shown[4] that the second hamiltonian structure of the generalized equations of KdV type[5] is the classical version of the extended conformal algebras, i.e., WN algebras[6]. Furthermore Polyakov[7] introduced a new class of WN algebras, denoted W (l) N , which was further investigated by Bershadsky[8], and it gives the poisson brackets of the second hamiltonian structure of the fractional KdV equations[9, 10, 11]. Another relation between the extended conformal algebras WN and the generalized KdV systems is that the classical bosonization rules for the WN algebras are given by the Miura map which is nothing but the relation between two different choices of gauge slice[12], and the free field representations for the W (l) N algebras can be also obtained by the same method[11]. More recently, an action was constructed that gives the KdV or mKdV equations as equations of motion[13], and their hamiltonian structures appear as poisson bracket structures derived from this action. Since the second hamiltonian structure of the KdV equation is given by the local poisson brackets, the corresponding symplectic form must contain a non-local operator, and it follows that the action has to be non-local in the KdV variable. However, one can write a local action[13] for the so called “Ur-KdV equation”, which is related to the KdV equation through the antipletic pair formalism of Wilson[14]. Such a local action constitutes of two parts; the kinetic term gives the evolution part of KdV equation that describes a free theory in an appropriate variable, and the potential term is written in terms of the infinite number of conserved quantities of KdV system. In this paper, we first rederive the KdV equation using zero curvature formulation with