Square-integrable solutions to a family of nonlinear schrِdinger equations from nonlinear quantum theory

We consider nonlinear Schrödinger equations which have been proposed as fundamental equations of nonlinear quantum theories. The equations are singular in that the wave function ψ appears in the denominator of rational expressions. To avoid the problem of zeros of ψ it is natural to make the ansatz ψ = eν. This ansatz, however, conflicts with the—physically motivated—requirement that the solutions ψ be square integrable. We show that this conflict can be resolved by considering an unusual function space whose definition involves the derivative ∇ν of ν. This function space turns out to be dense subset of L2 and the equations can be solved in the L2-sense (as desired) by first solving an evolutionary system for ∇ν and then transforming back to ψ.