Supersymmetric Representations and Integrable Fermionic Extensions of the Burgers and Boussinesq Equations

TL;DR

This paper develops new supersymmetric integrable systems, including fermionic extensions of classical equations like Burgers and Boussinesq, and analyzes their symmetries, conservation laws, and Hamiltonian structures.

Contribution

It introduces novel supersymmetric coupled systems and fermionic extensions of classical equations, with detailed symmetry and Hamiltonian analyses.

Findings

Existence of infinitely many higher symmetries demonstrated.

Construction of fermionic extensions related to associative algebra flows.

Discovery of a three-parametric supersymmetric system with unique properties.

Abstract

We construct new integrable coupled systems of N=1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian structures. A fermionic extension of the Burgers equation is related with the Burgers flows on associative algebras. A Gardner's deformation is found for the bosonic super-field dispersionless Boussinesq equation, and unusual properties of a recursion operator for its Hamiltonian symmetries are described. Also, we construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters.