Nonlinear canonical quantum system of collectively interacting particles via an exclusion-inclusion principle

Recently [G. Kaniadakis, Phys. Rev. A 55, 941 (1997)], we introduced a Schrödinger equation containing a complex nonlinearity W(ρ,j)+iW(ρ,j) which describes the collective interaction introduced by an exclusion-inclusion principle (EIP). The EIP does not affect W(ρ,j) and determines W(ρ,j) univocally. In the above reference W(ρ,j) was deduced by means of a stochastic quantization approach, in this way obtaining a noncanonical quantum system. In this work we introduce a family of nonlinearities W(ρ,j) generating a family of nonlinear canonical quantum systems, and derive their Lagrangian and the Hamiltonian functions and the evolution equations of the fields. We derive also the Ehrenfest relations and study the soliton properties. The shape of the soliton, propagating in the system obeying the EIP, can be obtained by solving a first-order ordinary differential equation. We show that, in the case of soliton solutions, by means of a unitary transformation, the EIP potential is equivalent to a real algebraic nonlinear potential proportional to κρ2/(1+κρ).