Linear problems and hierarchies of Painlevé equations

Abstract

In this paper, we show that the expansion of linear problems of the Painlevé equation in powers of the spectral variable can be used to derive hierarchies of ordinary differential equations. We applied this approach to linear problems of the first, second, third and fourth Painlevé equations. We derived a new hierarchy of the third Painlevé equation and rederived known hierarchies of the other equations. Moreover some special solutions of the hierarchies of the second, third and fourth Painlevé equations are also given.

Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy

“…The rogue wave of order k = 0 coincides with the background solution. Indeed, if k = 0, then the solution of Riemann-Hilbert Problem 1 is In verifying the jump condition (11) one should make use of the fact that the first three factors appearing on the second line of the right-hand side in (15) combine, perhaps despite appearances, to form an entire function U(λ; x, t) of λ: noting that analyticity follows because sin(θ)/θ and cos(θ) are even in θ and hence entire functions of θ 2 = (λ 2 + 1)(x + λt) 2 . Applying the formula (14) for k = 0 then gives (17) In [1], the conditions of Riemann-Hilbert Problem 1 were translated into a finite-dimensional linear algebra problem via a suitable rational ansatz for the matrix M (k) (λ; x, t)E(λ) −1 in the exterior domain that builds in poles of order n at λ = ±i (only visible upon analytic continuation into the interior domain through Σ • ).…”

Random Matrix Ensembles with Singularities and a Hierarchy of Painlevé III Equations

“…. defined by R p j (x) p ℓ (x)|x| 2α e − dx = h j (n, k, α)δ jℓ 23[1, Appendix B] and[17, Exercise 4.4]): p 2j (u; 2n, k, α) = p j (u 2 ; n, k, α − 1/2), (7.27) p 2j+1 (u; 2n, k, α) = up j (u 2 ; n, k, α + 1/2).…”

Bäcklund Transformations for First and Second Painlevé Hierarchies

“…In the present article, we generalize known Bäcklund transformations of the first and second Painlevé equations to the first and second Painlevé hierarchies given in [6,11]. We give a Bäcklund transformation between the considered first Painlevé hierarchy and a new hierarchy of Painlevé-type equations.…”

“…Moreover its members may define new transcendental functions. The PI hierarchy (2.1) can be written in the following form [11]…”