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Integral Equations and Exact Solutions for the Fourth Painleve Equation

A. P. Bassom, P. A. Clarkson, A. C. Hicks and J. B. McLeod
Proceedings: Mathematical and Physical Sciences
Vol. 437, No. 1899 (Apr. 8, 1992), pp. 1-24
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52049
Page Count: 24
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Integral Equations and Exact Solutions for the Fourth Painleve Equation
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Abstract

We consider a special case of the fourth Painleve equation given by d2η /dξ 2 = 3η 5 + 2ξ η 3 + (1/4ξ 2-ν -1/2)η , (1) with ν a parameter, and seek solutions η (ξ;ν) satisfying the boundary condition η (∞) = 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 η (ξ;ν) and η (ξ;ν +1).

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