A new class of Euler equation on the dual of the N = 1 extended Neveu-Schwarz algebra

Abstract

Let G be the N = 1 extended Neveu-Schwarz algebra and Greg* its regular dual. In this paper, we will study a super-Euler system with seven parameters (s1, s2, c1, …, c5) associated with Greg*. We will show that the super-Euler system is (1) local bi-superbihamiltonian if s1=14c1 and s2=12c2; (2) supersymmetric if s1 = c1 and s2 = c2; (3) local bi-superbihamiltonian and supersymmetric if s1 = c1 = 0 and s2 = c2 = 0. By choosing different parameters, we could obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the Ito equation, the 2-component Camassa-Holm equation, the 2-component Hunter-Saxton equation, and, especially, the Whitham-Broer-Kaup dispersive water-wave system.