On the supersymmetric nonlinear evolution equations

Supersymmetrization of a nonlinear evolution equation in which the bosonic equation is independent of the fermionic variable and the system is linear in fermionic field goes by the name B-supersymmetrization. This special type of supersymmetrization plays a role in superstring theory. We provide B-supersymmetric extension of a number of quasilinear and fully nonlinear evolution equations and find that the supersymmetric system follows from the usual action principle while the bosonic and fermionic equations are individually non Lagrangian in the field variable. We point out that B-supersymmetrization can also be realized using a generalized Noetherian symmetry such that the resulting set of Lagrangian symmetries coincides with symmetries of the bosonic field equations. This observation provides a basis to associate the bosonic and fermionic fields with the terms of bright and dark solitons. The interpretation sought by us has its origin in the classic work of Bateman who introduced a reverse-time system with negative friction to bring the linear dissipative systems within the framework of variational principle.