The focus of this paper is on the probability, Eβ(Q; J), that a set J consisting of a finite union of intervals contains no eigenvalues for the finite N Gaussian Orthogonal (β = 1) and Gaussian Symplectic (β = 4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the c...

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 2 and 2α = (2μ−1)2 with arbitrary μ, 2μ 6∈ Z. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in th...

The theory of solitary waves has attracted much interest in recent years for treatment of PDEs describing nonlinear and evolution concepts. Nonlinear phenomena appear in many areas of scientific fields such as solid state physics, plasma physics, fluid dynamics, mathematical biology and chemical kinetics. The nonlinear problems are characterized by dispersive effects, dissipative effects, conve...

The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be ...

The Hénon–Heiles system in the general form are studied. In a noninte-grable case new solution has been found as formal Laurent series, depending on three parameters. This asymptotic solution with some values of two parameters coincides with known exact solution. Let us consider a three-dimensional galaxy with an axial symmetric and time-independent potential function. The equations of galactic...

The first Painlevé equation is expressed as a vector field on the three dimensional weighted projective space CP (3, 2, 4, 5). With the aid of the dynamical systems theory and the orbifold structure of CP (3, 2, 4, 5), a simple proof of the Painlevé property and the construction of the space of initial conditions are given. In particular, Painlevé’s transformation is geometrically derived, whic...

We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation d 2w dz2 = 2w3 + zw + α, where α < 0 is a constant. Shadow kinks are sign changing solutions which satisfyw(z) ∼ −√−z/2 as z → −∞ and w(z) ∼ −z as z → ∞.These solutions play a critical role in the prediction of a new class of topological defects, one dimensional shadow kinks and two dimensional ...