The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.

Abstract. We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pureimpulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. Subject to suitable conditions of a general nature, we analyze the fluxon condensate solution approximating the given initial data for small time near points where the initial data crosses the separatrix of the phase portrait of the simple pendulum. We show that the solution is locally constructed as a universal curvilinear grid of superluminal kinks and grazing collisions thereof, with the grid curves being determined from rational solutions of the Painlevé-II system.

The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrödinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock front at large times.

Abstract. This paper is a continuation of our analysis, begun in [7], of the rational solutions of the inhomogeneous Painlevé-II equation and associated rational solutions of the homogeneous coupled Painlevé-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behavior. Our results display both a trigonometric degeneration of the rational Painlevé-II functions and also a degeneration to the tritronquée solution of the Painlevé-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann-Hilbert representation of the rational Painlevé-II functions, and supplies leading-order formulae as well as error estimates.

We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. We show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases. Contents 1 arXiv:1103.0061v1 [math-ph] 1 Mar 2011Using ω = v p k together with the nonlinear dispersion relation in the form (1.22) shows that k and ω may be eliminated in favor of v p and E:where σ = ±1 is an arbitrary sign whose role is to select different branches of the dispersion relation. Therefore, the variational modulation equation (1.23) becomesand the conservation of waves equation (1.11) becomesThe proof of this proposition is a rather straightforward application of Fubini's Theorem and is given in Appendix A.We will require that the WKB phase integral have certain analyticity properties to be outlined in Proposition 1.2 below. We now make an assumption on G that will be sufficient to establish Proposition 1.2 and that can easily be checked for a given G:Assumption 1.4. The function G is strictly increasing and real-analytic at each x > 0, and the positive and real-analytic functioncan be analytically continued to neighborhoods of m = 0 and m = G(0) 2 , with G (0) > 0 and G (G(0) 2 ) > 0.We point out that the class of functions G (m) satisfying Assumption 1.4 obviously parametrizes a corresponding class of admissible functions G(x) by simply viewing (1.48) as an equation to be solved for x = G −1