Fermionic Korteweg-De Vries Equation

The first extension of this kind is Kupershmidt’s [1,2] superKdV

(1)

where u is a bosonic and ϕ a fermionic field. The second simplest extension is Manin–Radul N=1 superKdV [3],

(2)

where χ = θ u + ϕ is a fermionic superfield and D = θ ∂/∂x + ∂/∂θ. Eq. (2) appears as a reduction of the superextension [3] of the KP hierarchy .

Unlike (2), (1) is not invariant [4] under Δ_η u =ηϕ _x , Δ_ηϕ = η u, where η is a Grassman variable, so it is not supersymmetric in the same sense as (2).

Both of (1) and (2) possess Lax representations, compatible Hamiltonian structures, infinitely many conservation laws, and other common features of integrable systems ; Bihamiltonian reduction and sKdVs, and the surveys [2,4] for more details on (1), (2), N = m superKdVs with m > 1, and other supersymmetric integrable systems .

Note that integrable systemsinvolving fermionic degrees of freedom are a particular case of...