On gauge transformations of B\"acklund type and higher order nonlinear Schr\"odinger 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 B\"acklund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schr\"odinger equation results in higher order partial differential equations. As an example, we derive a general family of 6th-order nonlinear Schr\"odinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current ${\bf \sigma}=\rho {\bf \nabla}\triangle \ln \rho$, where $\rho=\bar\psi \psi$. We derive gauge invariant quantities, and characterize the subclass of the 6th-order equations that is gauge equivalent to the free Schr\"odinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz.