Narrow Results

We apply Painlevé test to the most general variable coefficient nonlinear Schrödinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like group-theoretical approach and the Lax pairs technique of the soliton theory. We presented some exact solutions based on point transformations relating analytic solutions of VCNLS equations for specific choices of the coefficients to those of the standard integrable NLS equation.

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the examples the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.