N=2 Super Boussinesq Hierarchy: Lax Pairs and Conservation Laws

TL;DR

This paper investigates the integrability of the N=2 super Boussinesq hierarchy, identifying specific parameter values where it admits conserved quantities and Lax pairs, thus establishing its integrable structure.

Contribution

It demonstrates that the N=2 super Boussinesq equations are integrable only for three specific parameter values, providing Lax pairs and bi-Hamiltonian structures for some cases.

Findings

Integrability occurs only at three parameter values: -2, -1/2, 5/2.

Lax pair formulation exists for alpha = -1/2.

The case alpha = -2 is bi-Hamiltonian.

Abstract

We study the integrability properties of the one-parameter family of $N=2$ super Boussinesq equations obtained earlier by two of us (E.I. \& S.K., Phys. Lett. B 291 (1992) 63) as a hamiltonian flow on the $N=2$ super-$W_3$ algebra. We show that it admits nontrivial higher order conserved quantities and hence gives rise to integrable hierarchies only for three values of the involved parameter, $\alpha=-2,\;-1/2,\;5/2$. We find that for the case $\alpha = -1/2$ there exists a Lax pair formulation in terms of local $N=2$ pseudo-differential operators, while for $\alpha = -2$ the associated equation turns out to be bi-hamiltonian.