On the Integrable Hierarchies Associated With N=2 Super WnW_n Algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with N=2 super $W_n$ algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulted hierarchy contains the N=2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric Boussinesq hierarchies.Comment: 11 pages, AMS-LaTex, to appear in Phys. Lett.

Two-Matrix String Model as Constrained (2+1)-Dimensional Integrable System

We show that the 2-matrix string model corresponds to a coupled system of $2+1$-dimensional KP and modified KP (\KPm) integrable equations subject to a specific ``symmetry'' constraint. The latter together with the Miura-Konopelchenko map for \KPm are the continuum incarnation of the matrix string equation. The \KPm Miura and B\"{a}cklund transformations are natural consequences of the underlying lattice structure. The constrained \KPm system is equivalent to a $1+1$-dimensional generalized KP-KdV hierarchy related to graded ${\bf SL(3,1)}$. We provide an explicit representation of this hierarchy, including the associated ${\bf W(2,1)}$-algebra of the second Hamiltonian structure, in terms of free currents.Comment: 12+1 pgs., LaTeX, preprint: BGU-94 / 15 / June-PH, UICHEP-TH/94-

On the constrained KP hierarchy

An explanation for the so-called constrained hierarhies is presented by linking them with the symmetries of the KP hierarchy. While the existence of ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP hierarchy to the KdV hierarchies, the existence of additional symmetries allows to reduce KP to the constrained KP.Comment: 7pp, LaTe

Compatible Poisson Structures of Toda Type Discrete Hierarchy

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.Comment: 22 pages, LaTeX, v3: Minor change

Virasoro Symmetry of Constrained KP Hierarchies

Additional non-isospectral symmetries are formulated for the constrained Kadomtsev-Petviashvili (\cKP) integrable hierarchies. The problem of compatibility of additional symmetries with the underlying constraints is solved explicitly for the Virasoro part of the additional symmetry through appropriate modification of the standard additional-symmetry flows for the general (unconstrained) KP hierarchy. We also discuss the special case of \cKP --truncated KP hierarchies, obtained as Darboux-B\"{a}cklund orbits of initial purely differential Lax operators. The latter give rise to Toda-lattice-like structures relevant for discrete (multi-)matrix models. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-B\"{a}cklund transformations of \cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models.Comment: LaTeX, 11 pg

A 2 - Component or N=2 Supersymmetric Camassa - Holm Equation

The extended N=2 supersymmetric Camasa - Holm equation is presented. It is accomplishe by formulation the supersymmeytric version of the Fuchssteiner method. In this framework we use two supersymmetric recursion operators of the N=2, $\alpha=-2,4$ Korteweg - de Vries equation and constructed two different version of the supersymmetric Camassa - Holm equation. The bosonic sector of N=2, $\alpha=4$ supersymmetric Camassa - Holm equation contains two component generalization of this equation considered by Chen, Liu and Zhang and as a special case two component generalized Hunter - Saxton equation considered by Aratyn, Gomes and Zimerman, As a byproduct of our analysis we defined the N=2 supersymmetric Hunter - Saxton equation. The bihamiltonian structure is constructed for the supersymmetric N=2, $\alpha=4$ Camassa - Holm equation.Comment: 9 pages, Latex,corrected typo

On symmetries of KdV-like evolution equations

The $x$-dependence of the symmetries of (1+1)-dimensional scalar translationally invariant evolution equations is described. The sufficient condition of (quasi)polynomiality in time $t$ of the symmetries of evolution equations with constant separant is found. The general form of time dependence of the symmetries of KdV-like non-linearizable evolution equations is presented.Comment: LaTeX, 8 pages, no figures, very minor change

Flat Pencils of Symplectic Connections and Hamiltonian Operators of Degree 2

Bi-Hamiltonian structures involving Hamiltonian operators of degree 2 are studied. Firstly, pairs of degree 2 operators are considered in terms of an algebra structure on the space of 1-forms, related to so-called Fermionic Novikov algebras. Then, degree 2 operators are considered as deformations of hydrodynamic type Poisson brackets.Comment: 20 page

Tri-hamiltonian vector fields, spectral curves and separation coordinates

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P_0, P_1, P_2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P_1 - \lambda P_0) and (P_2 - \mu P_0); (ii) a suitable set of vector fields, preserving P_0 but transversal to its symplectic leaves. The frameworks is applied to Lax equations with spectral parameter, for which not only it unifies the separation techniques of Sklyanin and of Magri, but also provides a more efficient ``inverse'' procedure not involving the extraction of roots.Comment: 49 pages Section on reduction revisite

Integrable Extensions of N=2 Supersymmetric KdV Hierarchy Associated with the Nonuniqueness of the Roots of the Lax operator

We preesent a new supersymmetric integrable extensions of the a=4,N=2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generate new hierarchy of integrable equations, for which we investigate the hamiltonian structure. In special case our system describes the interaction of the KdV equation with the two MKdV equations.Comment: 9 pages Latex,e-mail [email protected]