N=2 Super - $W_{3}$ Algebra and N=2 Super Boussinesq Equations

TL;DR

This paper develops a geometric framework for N=2 super W_3 algebra and derives new N=2 super Boussinesq equations, exploring their integrability and relations via super Miura maps.

Contribution

It introduces a covariant reduction method to derive the most general N=2 super Boussinesq equations and their modifications, linking them through super Miura maps.

Findings

Derived the general N=2 super Boussinesq equation.

Established super Miura maps relating different systems.

Discussed integrability and Hamiltonian structures of the equations.

Abstract

We study classical $N=2$ super-$W_3$ algebra and its interplay with $N=2$ supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs - covariant reduction approach. These techniques have been previously applied by us in the bosonic $W_3$ case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general $N=2$ super Boussinesq equation and two kinds of the modified $N=2$ super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear $N=2$ super-$W_3^{\infty}$ symmetry associated with $N=2$ super-$W_3$. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second hamiltonian structure.