On Integrable Doebner–Goldin Equations

We suggest a method for integrating sub-families of a family of nonlinear Schrödinger equations proposed by H.-D. Doebner and G.A. Goldin in the 1+1 dimensional case which have exceptional Lie symmetries. Since the method of integration involves non-local transformations of dependent and independent variables, general solutions obtained include implicitly determined functions. By properly specifying one of the arbitrary functions contained in these solutions, we obtain broad classes of explicit square integrable solutions. The physical significance and some analytical properties of the solutions obtained are briefly discussed.