Pole-free solutions of the first Painlevé hierarchy and non-generic critical behavior for the KdV equation

Abstract

We establish the existence of real pole-free solutions to all even members of the Painlevé I hierarchy. We also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the KdV equation in the small dispersion limit. This was understood in the case of a generic critical point, and we generalize it here to the case of non-generic critical points.

Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation

“…In this case, we expect that Φ n → Φ 0 for a new function Φ 0 . It is relatively simple to write a RHP that should be satisfied by this Φ 0 , and we expect it to be related to the KdV hierarchy [28] but with nonstandard initial data. It would be interesting to see if the particular solutions obtained this way reduce to integro-differential hierarchies of Painlevé equations, in the same spirit of the recent works [23,46].…”

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

“…This hierarchy of ODE is known [11], [12] as the hierarchy of massive (2n + 1, 2) string equations. It is also referred to as the P n 1 -hierarchy of the first Painlevé equation (see, for example, [12], [13]).…”

“…Many publications in the last quarter century were devoted to studying various properties of simultaneous solutions of ODE and KdV (first of all, the Gurevich-Pitaevskii special solution); see [29]- [39] and references therein in addition to those given above. Other representatives of the hierarchy of simultaneous solutions of ( 16) and ( 4) are also related to the description of formation of dispersive shock waves in degenerate cases [13], [29].…”

From gap probabilities in random matrix theory to eigenvalue expansions

“…Remark 1.15. The particular scaling chosen in (1.18) differs slightly from [16] and also [13]. For instance Ψ(ζ; s, t 1 ) in [13], Section 2, connects to Φ(ζ; x, τ ) in RHP 1.14 via…”

“…The precise expressions for A ( ) z and B ( ) z can be found in, say, [13] but they will not be relevant for us, we only use the existence of a real-valued, pole-free solution of (1.19) for x,  t Î , see again [13]. Equivalently we only use solvability of RHP 1.14 for x,  t Î .…”