On the application of solutions of the fourth Painlevé equation to various physically motivated nonlinear partial differential equations

Abstract

Recently significant developments have been made in the understanding of the theory and solutions of the fourth Painlev\'e equation given by \begin{equation} ww'' = \tfrac 12(w')^2+\tfrac 32w^4 + 4zw^3 + 2(z^2-\alpha)w^2 + \beta, \tag{1} \end{equation} where $\alpha$ and $\beta$ are arbitrary constants. All exact solutions of (1) are thought to belong to one of three solution hierarchies. In two of these hierarchies solutions may be determined in terms of the complementary error and parabolic cylinder functions whilst the third hierarchy contains rational solutions which may be expressed as ratios of polynomials in $z$. It is known that the fourth Painlevé equation (1) can arise as a symmetry reduction of a number of significant partial differential equations and here we consider the application of solutions of (1) to the nonlinear Schrödinger, potential nonlinear Schrödinger, spherical Boussinesq and $2+1$-dimensional dispersive long wave equations. Consequently we generate a number of new exact solutions of these equations. The fourth Painlevé equation (1) is also related to problems that arise in the consideration of two-dimensional quantum gravity and this aspect is also discussed