A 2-component or N=2N=2 supersymmetric Camassa–Holm equation

The extended N=2N=2 supersymmetric Camassa–Holm equation is presented. It is accomplished by formulating the supersymmetric version of the Fuchssteiner method. In this framework we use two supersymmetric recursion operators of the N=2N=2, α=−2,4α=−2,4 Korteweg–de Vries equation and construct two different versions of the supersymmetric Camassa–Holm equation. The bosonic sector of N=2N=2, α=4α=4 supersymmetric Camassa–Holm equation contains the two component generalization of this equation proposed by Chen, Liu and Zhang and as a special case the two component generalized Hunter–Saxton equation considered by Aratyn, Gomes and Zimerman. As a byproduct of our analysis we defined the N=2N=2 supersymmetric Hunter–Saxton equation. The bihamiltonian structure is constructed for the supersymmetric N=2N=2, α=4α=4 Camassa–Holm equation.