Super Hamiltonian Systems Research Articles

Supersymmetric Integrable Hamiltonian Systems, Conformal Lie Superalgebras K(1, N = 1, 2, 3), and Their Factorized Semi-Supersymmetric Generalizations

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional N=1,2,3- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable N=2,3-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra K(1|3) and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the N=1,2,3-supercircle.

Classical Affine W-Superalgebras via Generalized Drinfeld–Sokolov Reductions and Related Integrable Systems

The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the Poisson vertex algebras theory. A W-superalgebra with certain properties can be understood as an underlying differential superalgebra of a series of integrable super-Hamiltonian systems.

A hierarchy of super AKNS hierarchy related to Lie superalgebra sl(2|1) and a finite dimensional super Hamiltonian system

A hierarchy of super AKNS equations associated with a [Formula: see text] matrix-valued spectral problem is derived. It is shown that each equation in the hierarchy is bi-super Hamiltonian. Moreover, a new finite dimensional super Hamiltonian system (FDSHS), together with its Lax representation, [Formula: see text]-matrix and conversed integrals of motion, is obtained from the spectral problem by binary nonlinearization.

Two-Component Super AKNS Equations and Their Finite-Dimensional Integrable Super Hamiltonian System

Starting from a matrix Lie superalgebra, two-component super AKNS system is constructed. By making use of monononlinearization technique of Lax pairs, we find that the obtained two-component super AKNS system is a finite-dimensional integrable super Hamiltonian system. And its Lax representation andr-matrix are also given in this paper.

Binary nonlinearization of the super AKNS system under an implicit symmetry constraint

For the super AKNS system, an implicit symmetry constraint between the potentials and the eigenfunctions is proposed. After introducing some new variables to explicitly express potentials, the super AKNS system is decomposed into two compatible finite-dimensional super systems (x-part and tn-part). Furthermore, we show that the obtained super systems are integrable super Hamiltonian systems in the supersymmetry manifold .

Hamiltonian and super-Hamiltonian systems of a hierarchy of soliton equations

By considering an isospectral eigenvalue problem, a hierarchy of soliton equations are derived. Two types of extensions are presented by enlarging the associated spectral problem. With the aid of generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures for the integrable extensions are established.