Bilinear approach to the supersymmetric Gardner equation

Abstract

We study a supersymmetric version of the Gardner equation (both focusing and defocusing) using the superbilinear formalism. This equation is new and cannot be obtained from the supersymmetric modified Korteweg–de Vries equation with a nonzero boundary condition. We construct supersymmetric solitons and then by passing to the long-wave limit in the focusing case obtain rational nonsingular solutions. We also discuss the supersymmetric version of the defocusing equation and the dynamics of its solutions.