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

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.

1+19 pages, 3 figures; v2: reference added; v3: more references added, published in LMP

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Cited by in corpus (12)

• Global existence for the generalized two-component Hunter-Saxton system
• A 2-component $\mu$-Hunter-Saxton equation
• The curvature of semidirect product groups associated with two-component Hunter-Saxton systems
• Euler Equations Related to the Generalized Neveu-Schwarz Algebra
• On a two-component $\pi$-Camassa--Holm system
• A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue
• A note on multi-dimensional Camassa-Holm type systems on the torus
• The transformations between N=2 supersymmetric KdV and HD equations
• On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations
• Global estimates and blow-up criteria for the generalized Hunter-Saxton system
• On initial boundary value problems for variants of the Hunter-Saxton equation
• Geometric numerical integrators for Hunter-Saxton-like equations