The Supersymmetric Two Boson Hierarchies

TL;DR

This paper constructs the most general integrable supersymmetric two boson system, providing its Lax representations, Hamiltonian structures, and connections to known integrable models like the supersymmetric nonlinear Schrödinger and KdV equations.

Contribution

It introduces the most general supersymmetric two boson hierarchy with explicit Lax operators and demonstrates its relation to established integrable systems.

Findings

Derived the Lax operator and nonstandard Lax representation.

Showed reduction to supersymmetric nonlinear Schrödinger equation.

Established three local Hamiltonian structures and embedding of supersymmetric KdV.

Abstract

We construct the most general supersymmetric two boson system that is integrable. We obtain the Lax operator and the nonstandard Lax representation for this system. We show that, under appropriate redefinition of variables, this reduces to the supersymmetric nonlinear Schr\"odinger equation without any arbitrary parameter which is known to be integrable. We show that this supersymmetric system has three local Hamiltonian structures just like the bosonic counterpart and we show how the supersymmetric KdV equation can be embedded into this system.