PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation

We discuss several PT-symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT-symmetric deformations of the supersymmetric Korteweg-de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace valued superfields containing bosonic and fermionic fields, breaking however the supersymmetry invariance. Nonetheless, we also find extensions, which may be viewed as new supersymmetric Korteweg-de Vries equation. Moreover, we show that these deformations allow for a non-Hermitian Hamiltonian formulation and construct three charges associated to the corresponding flow.Comment: 10 page

The Integrability of New Two-Component KdV Equation

We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel'd and Sokolov and rediscovered by Sakovich and Foursov. Connection of this equation with the supersymmetric Kadomtsev-Petviashvilli-Radul-Manin hierarchy is presented. For this new supersymmetric equation the Lax representation and odd Hamiltonian structure is given

Supersymmetric Harry Dym Type Equations

A supersymmetric version is proposed for the well known Harry Dym system. A general class super Lax operator which leads to consistent equations is considered.Comment: 4 pages, latex, no figure

(Non)local Hamiltonian and symplectic structures, recursions, and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation

Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000], we accomplish an extensive study of the N=1 supersymmetric Korteweg-de Vries equation. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. We stress that the main point of the paper is not just the results on super-KdV equation itself, but merely exposition of the efficiency of the geometrical approach and of the computational algorithms based on it.Comment: 16 pages, AMS-LaTeX, Xy-pic, dvi-file to be processed by dvips. v2: nonessential improvements of exposition, title change