Equivalence transformations and differential invariants of a generalized nonlinear Schr\"{o}dinger equation

By using the Lie's invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear Schr\"{o}dinger equations with variable coefficients. Starting from the equivalence generators we construct the differential invariants of order one. We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schr\"{o}dinger equations which can be mapped, by means of an equivalence transformation, to the well known cubic Schr\"{o}dinger equation. We also provide the explicit form of the transformation.