On preliminary symmetry classification of nonlinear Schrِdinger equations with some applications to Doebner-Goldin models

We perform classification of a class of one-dimensional nonlinear Schrِdinger equations whose symmetry groups have dimensions n = 1, 2, 3. Next, from so constructed classes of invariant equations we select those nonlinear Schrِdinger equations which are invariant with respect to the Galilei group and its natural extensions. The results obtained are applied for the symmetry classification of complex Galilei-invariant Doebner-Goldin models.