On gauge transformations of Baecklund type and higher order nonlinear Schroedinger equations

We introduce a new, more general type of nonlinear gauge transformation in nonrelativistic quantum mechanics that involves derivatives of the wave function and belongs to the class of Baecklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schroedinger equation results in higher order partial differential equations. As an example, we derive a general family of sixth-order nonlinear Schroedinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current σ=ρ∇Δ ln ρ, where ρ=ψ(bar sign)ψ. We derive gauge invariant quantities, and characterize the subclass of the sixth-order equations that is gauge equivalent to the free Schroedinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz