N=2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations

TL;DR

This paper introduces the N=2 supercomplexification method for integrable equations, specifically applying it to the Korteweg-de Vries, Sawada-Kotera, and Kaup-Kupershmidt equations, revealing new structures and conservation laws.

Contribution

It defines and investigates N=2 supercomplex versions of these equations, highlighting their odd bi-Hamiltonian structures and superfermionic conservation laws.

Findings

Presence of odd Hamiltonian structures

Existence of superfermionic conserved currents

Construction of Lax representations for the equations

Abstract

The supercomplexification is a special method of N=2 supersymmetrization of the integrable equations in which the bosonic sector could be reduced to the complex version of these equations. The N=2 supercomplex Korteweg de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd hamiltonian structures and superfermionic conservation laws. The odd bi-hamiltonian structure, Lax representation and superfermionic conserved currents for new N=2 supersymmetric Korteweg de Vries equation and for Sawada-Kotera, are given. The N=2 supercomplex Kaup-Kupershmidt equation is defined for which the odd bi-hamiltonian structure is presented with its superfermionic conserved currents.