On the solvability of Painleve I, III and V

Abstract

As rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ODEs was introduced in 1992 by Fokas and Zhou, and was used to investigate Painleve II and Painleve IV equations. Here the authors apply this methodology to Painleve I, III, and V equations. They show that the Cauchy problems for these equations admit in general global solutions, meromorphic in t. Furthermore, for special relations among the monodromy data and for t on Stokes lines, these solutions are bounded for finite t. In connection with Painleve I they note that the usual Lax pair gives rise to monodromy data some of which depend nonlinearly on the unknown solution of Painleve I. This problem is bypassed here by introducing a new Lax pair for which all the monodromy data are constant.

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…So, by an argument similar to Section 3.7, we have 17) where µ = 1 for α > 0, and µ = α + 1 for −1 < α < 0. This completes the nonlinear steepest descend analysis of Ψ(ζ, s) as s → 0 + .…”

“…The vanishing lemma states that the null space is trivial, which implies that the singular integral equation (and thus Ψ 0 ) is solvable as a result of the Fredholm alternative theorem. More details can be found in [22,Proposition 2.4]; see also [10,12,15,17] for standard methods connecting RH problems with integral equations. Now we have the following solvability result:…”

The KdV Hierarchy: Universality and a Painlevé Transcendent

“…The relevant solution is real and has the asymptotic behavior 18) for any fixed T ∈ R, and has no poles for real values of X and T [5,34,35]. It is also remarkable that U (X, T ) is an exact solution to the KdV equation normalized as…”

“…Remark 2.1 Using the vanishing lemma approach developed in [19,18,17], one can show that the RH problem for M is solvable for any value of x, t ∈ R, ǫ > 0 if r 0 has sufficient regularity and sufficient decay at −∞ and if |r 0 (λ)| < 1 for λ < 0. The solvability of the RH problem can be used to prove that the Cauchy problem for equation (1.2) is solvable for initial data in a suitable space following the proofs in [41].…”

The Asymptotic Behavior of the Solution of Boundary Value Problems for the sine-Gordon Equation on a Finite Interval

“…The Riemann-Hilbert problem with respect to these variables is identical to the RiemannHilbert problems characterizing the solution of the so-called Painlevé III ODE, which is analyzed in [6]. With respect to the variable λ, the kernel of equation (3.13) is…”