Gelfand-Dikii Brackets for Nonstandard Supersymmetric Systems

TL;DR

This paper demonstrates how to convert nonstandard supersymmetric Lax equations into standard form, enabling the definition of Gelfand-Dikii brackets and analysis of Hamiltonian structures for supersymmetric integrable systems.

Contribution

It introduces a method to express nonstandard Lax equations as standard ones, facilitating the definition of Gelfand-Dikii brackets for supersymmetric systems.

Findings

Established a way to express nonstandard Lax equations as standard equations.

Defined Gelfand-Dikii brackets for nonstandard supersymmetric equations.

Explicitly derived Hamiltonian structures for supersymmetric Two Boson system.

Abstract

We show how a general nonstandard Lax equation (supersymmetric or otherwise) can be expressed as a standard Lax equation. This enables us to define the Gelfand-Dikii brackets for a nonstandard supersymmetric equation. We discuss the Hamiltonian structures for the nonstandard super KP system and work out explicitly the two Hamiltonian structures of the supersymmetric Two Boson system from this point of view.