On the N=2 Supersymmetric Camassa-Holm and Hunter-Saxton Equations

TL;DR

This paper explores N=2 supersymmetric extensions of the Camassa-Holm and Hunter-Saxton equations, revealing their geometric interpretations, bi-Hamiltonian structures, Lax pairs, and explicit solutions, including bounded traveling waves.

Contribution

It introduces supersymmetric extensions with geometric and algebraic structures, deriving Lax pairs and explicit solutions, advancing understanding of these integrable systems.

Findings

Supersymmetric extensions admit geometric interpretations as Euler equations.

Derived Lax pairs using bi-Hamiltonian formulation.

Obtained explicit solutions, including bounded traveling waves.

Abstract

We consider N=2 supersymmetric extensions of the Camassa-Holm and Hunter-Saxton equations. We show that they admit geometric interpretations as Euler equations on the superconformal algebra of contact vector fields on the 1|2-dimensional supercircle. We use the bi-Hamiltonian formulation to derive Lax pairs. Moreover, we present some simple examples of explicit solutions. As a by-product of our analysis we obtain a description of the bounded traveling-wave solutions for the two-component Hunter-Saxton equation.