On the N=2 supersymmetric Camassa-Holm and Hunter-Saxton equations

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.