List of Figures2.1 The complex λ-plane with three eigenvalues λ k in the upper half-plane, their complex conjugates, and the contours C, C * , and domains D, D * . The orientation in the figure is ω = +1. 4.1 Examples of contours C and C * appropriate for studying a semiclassical soliton ensemble in the limit N ↑ ∞.

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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.

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary‐order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher‐order “rogue wave” solutions in an inverse‐scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.

We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order k, we establish the existence of a limiting profile of the rogue wave in the large-k limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables -the rogue wave of infinite order -which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulae with the exact solution with the help of numerical methods for solving Riemann-Hilbert problems. In a certain transitional region for the asymptotics the near field limit function is described by a specific globally-defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.

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.

e-print archive: http://xxx.lanl.gov/math.PR/0112162 1208 BAIK, DEIFT, MCLAUGHLIN, MILLER, AND ZHOUIn this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.

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