Application of the isomonodromy deformation method to the fourth Painlevé equation

In this paper we study the fourth Painlevé equation and how the concept of isomonodromy may be used to elucidate properties of its solutions. This work is based on a Lax pair which is derived from an inverse scattering formalism for a derivative nonlinear Schrödinger system, which in turn possesses a symmetry reduction that reduces it to the fourth Painlevé equation. It is shown how the monodromy data of our Lax pair can be explicitly computed in a number of cases and the relationships between special solutions of the monodromy equations and particular integrals of the fourth Painlevé equation are discussed. We use a gauge transformation technique to derive Bäcklund transformations from our Lax pair and generalize the findings to examine particular solutions and Bäcklund transformations of a related nonlinear harmonic oscillator equation.