Integral Equations and Exact Solutions for the Fourth Painleve Equation
Abstract
We consider a special case of the fourth Painleve equation given by d^{2}eta /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).
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 April 1992
 DOI:
 10.1098/rspa.1992.0043
 Bibcode:
 1992RSPSA.437....1B