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

Double Extended Cubic Peakon Equation

The Hamiltonian structure for the supersymmetric $N=2$ Novikov equation is presented. The bosonic sector give us two-component generalization of the cubic peakon equation. The double extended: two-component and two-peakon Novikov equation is defined. The Bi-Hamiltonian structure for this extended system is constructed.Comment: the misprints are correcte

The Super W_3 Conformal Algebra and the Boussinesq Hierarchy

The bihamiltonian structure of the N=2 Supersymmetric Boussinesq equation is found. It is not reduced to the corresponding classical structure and hence it describes the pure supersymmetric effect. For the supersymmetric Boussinesq equation which contains the classical partner the Lax pair is given explicitly. Thus we prove the integrability of the equation.Comment: 12 page

Odd Bihamiltonian Structure of New Supersymmetric N=2,4 KdV And Odd SUSY Virasoro - Like Algebra

The general method of the supersymmetrization of the soliton equations with the odd (bi) hamiltonian structure is established. New version of the supersymmetric N=2,4 (Modified) Korteweg de Vries equation is given, as an example. The second odd Hamiltonian operator of the SUSY KdV equation generates the odd N=2,4 SUSY Virasoro - like algebra.Comment: 13 pages LaTe

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]

Extensions of the N=2 Supersymmetric a=-2 Boussinesq Hierarchy

We present two different Lax operators for a manifestly N=2 supersymmetric extension of "a=-2" Boussinesq hierarchy . The first is the supersymmetric generalization of the Lax operator of the Modified KdV equation. The second is the generalization of the supersymmetric Lax operator of the N=2 supersymmetric a=-2 KdV system. The gauge transformation of the first Lax operator provide the Miura link between the "small" N=4 supersymmetric conformal algebra and the supersymmetric $W_{3}$ algebra .Comment: LaTex, new references added, minor typos corrected, e-mail: [email protected]