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

Let G be the N = 1 extended Neveu-Schwarz algebra and G*(reg) its regular dual. In this paper, we will study a super-Euler system with seven parameters (s(1), s(2), c(1), . . ., c(5)) associated with G*(reg). We will show that the super-Euler system is (1) local bi-superbihamiltonian if s(1) = 1/4c(1) and s(2) = 1/2c(2); (2) supersymmetric if s(1) = c(1) and s(2) = c(2); (3) local bi-superbihamiltonian and supersymmetric if s(1) = c(1) = 0 and s(2) = c(2) = 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. Published by AIP Publishing.