A comparative analysis of Painlevé, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations

We derive the integrability conditions of nonautonomous nonlinear Schrödinger equations using the Lax pair and similarity transformation methods. We present a comparative analysis of these integrability conditions with those of the Painlevé method. We show that while the Painlevé 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 Painlevé method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schrödinger equations for two- and three-spacial dimensions.