Integral Equations and Exact Solutions for the Fourth Painleve Equation

We consider a special case of the fourth Painleve equation given by d²eta /dξ ² = 3eta ⁵ + 2ξ eta ³ + (1/4ξ ²-ν -1/2)eta , (1) with ν a parameter, and seek solutions eta (ξν) satisfying the boundary condition eta (∞) = 0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrodinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact `bound state' solutions of equation (1) for ν = n, a positive integer, using the integral equation representation, which decay exponentially as ξ -> ± ∞ and are the first example of such solutions for the Painleve equations. Additionally, using Backlund transformations for the fourth Painleve equation, we derive a nonlinear recurrence relation (commonly referred to as a Backlund transformation in the context of soliton equations) for equation (1) relating eta (ξν) and eta (ξν +1).