Four super integrable equations: nonlocal symmetries and applications

Abstract

By applying Hamiltonian operators to gradients of spectral parameters, nonlocal symmetries quadratically depending on eigenfunctions of linear spectral problems are constructed for super bi-Hamiltonian equations including a super modified Korteweg–de Vries (KdV) equation, a super K(−1, −2) equation, Kupershmidt’s super KdV equation and a super Ablowitz–Kaup–Newell–Segur system. In each example, the nonlocal symmetry is prolonged to an enlarged system, and generates a finite symmetry transformation. On this basis, a non-trivial solution, as well as a Bäcklund transformation, is established for the each super equation under consideration.