New complex-valued solutions of Painlevé IV: An application to the nonlinear Schrödinger equation

This paper is devoted to identifying some novel complex-valued solutions of the fourth Painlevé equation (P_IV). This equation possesses two parameters and past work has identified many exact solutions of P_IV for certain real values of this pair of constants. Two hierarchies of solutions are obtained that are distinguished by the fact that the associated parameters are complex valued. These solutions are applied to the nonlinear Schrödinger equation which enables a new similarity solution to be derived.