Integral equations and exact solutions for the fourth Painlevé equation

Abstract

We consider a special case of the fourth Painlevé equation given by d 2 ƞ / dξ 2 = 3 ƞ 5 + 2ξ ƞ 3 + (1/4ξ 2 - v - 1/2 ) ƞ , (1) with v a parameter, and seek solutions ƞ (ξ; v ) satisfying the boundary condition ƞ (∞)=0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (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 v = n , a positive integer, using the integral equation representation, which decay exponentially as ξ→ ± ∞ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating ƞ (ξ; v ) and ƞ (ξ; v + 1).

On Dirichlet two-point boundary value problems for the Ermakov–Painlevé IV equation

“…In [23], Bassom et al considered a class of Dirichlet boundary value problems for the avatar (2.1) of the integrable Painlevé IV equation but with the specialisation β = 0. In that paper, the concern was to isolate solutions Ω(α : x) constrained by the boundary condition Ω(∞) = 0.…”

Series Expansions of Painlevé Transcendents near the Point at Infinity

“…There have been numerous papers discussing two-parameter solutions in the studies of connection problems, isomonodromy deformations and others: for example, [3], [37], [20] for (I); [16] for (I), (II); [10], [19], [17], [11] for (II); [25], [22], [21], [24] for (III), (III Ã ), (III 0 ); [23], [4], [12] for (IV); [32], [2] for (V), (V 0 ) (see also [14], [7]). For pole-free asymptotic solutions among them, which oscillate along Stokes curves, our theorems provide their series expansions valid in the curved sectors.…”

Hirota bilinear approach to GUE, NLS, and Painlevé IV

“…Suppose τ (t 1 , t 2 ), q(t 1 , t 2 ) and r(t 1 , t 2 ) satisfy the differential Eqs. (13), (18), and the similarity conditions (29), (32), (33). Then one can introduce F (s) as…”

“…We also consider the asymptotic behavior of F 2,N (s) as s → ±∞, and its relation to the "ClarksonMcLeod solution" [18][19][20] to the Painlevé IV equation. This paper is dedicated to the memory of Professor Ryogo Hirota, who introduced his bilinear method that is a powerful tool for obtaining a wide class of exact solutions of nonlinear integrable equations, such as the NLS equation and the Painlevé IV equation.…”