A comparative analysis of Painlev\'e, Lax Pair, and Similarity Transformation methods in obtaining the integrability conditions of nonlinear Schr\"odinger equations

We derive the integrability conditions of nonautonomous nonlinear Schr$\ddot{\mathrm{o}}$dinger equations using the Lax Pair and Similarity Transformation methods. We present a comparative analysis of these integrability conditions with those of the Painlev$\acute{\mathrm{e}}$ method. We show that while the Painlev$\acute{\mathrm{e}}$ integrability conditions restrict the dispersion, nonlinearity, and dissipation/gain coefficients to be space-independent and the external potential to be only a quadratic function of position, the Lax Pair and the Similarity Transformation methods allow for space-dependent coefficients and an external potential that is not restricted to the quadratic form. The integrability conditions of the Painlev$\acute{\mathrm{e}}$ method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schr$\ddot{\mathrm{o}}$dinger equations for two- and three-spacial dimensions.