On gauge transformations of Bäcklund type and higher order nonlinear Schrödinger equations

Abstract

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