Table of contents

The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0, 0) to (1, 1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulae. The derivation of these formulae is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the Kardar–Parisi–Zhang universality class. Following works of Baik and Rains, and Prähofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlevé II transcendent plays a prominent role.

We endow the nonlinear degenerate parabolic equation used to describe propagation of thermal waves in plasma or in a porous medium, with a mechanism for flux saturation intended to correct the nonphysical gradient-flux relations at high gradients. We study both analytically and numerically the resulting equation: u_t = [uⁿQ(g(u)_x)]_x, n>0, where Q is a bounded increasing function. This model reveals that for n>1 the motion of the front is controlled by the saturation mechanism and instead of the typical infinite gradients resulting from the linear flux-gradients relations, Q∼u_x, we obtain a sharp, shock-like front, typically associated with nonlinear hyperbolic phenomena. We prove that if the initial support is compact, independently of the smoothness of the initial datum inside the support, a sharp front discontinuity forms in a finite time, and until then the front does not expand.

We consider the question of finding a periodic solution for the planar Newtonian N-body problem with equal masses, where each body is travelling along the same closed path. We provide a computer-assisted proof for the following facts: the local uniqueness and the convexity of the Chenciner and Montgomery Eight, the existence (and the local uniqueness) for Gerver's Super-Eight for 4-bodies and a doubly symmetric linear chain for 6-bodies.

The τ-function theory of Painlevé systems is used to derive recurrences in the rank n of certain random matrix averages over U(n). These recurrences involve auxiliary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as n varies, and demonstrating convergence to the value of the appropriate limiting distribution.

Burgers' model for turbulence consists of a PDE coupled with a nonlocal ODE. We study the behaviour, backward in time, of its solutions, which turns out to be quite different from the behaviour of the solutions of the space periodic two-dimensional Navier–Stokes equations (Constantin P, Foias C, Kukavica I and Majda A 1997 Dirichlet quotients and 2-D periodic Navier–Stokes equations J. Math. Pure Appl. 76 125–53) and space periodic one-dimensional Kuramoto–Sivashinsky equation (Kukavica I 1992 On the behavior of the solutions of the Kuramoto–Sivashinsky equations for negative time J. Math. Anal. Appl. 166 601–6).

The global regularity for the two- and three-dimensional Kuramoto–Sivashinsky equations is one of the major open questions in nonlinear analysis. Inspired by this question, we introduce in this paper a family of hyper-viscous Hamilton–Jacobi-like equations parametrized by the exponent in the nonlinear term, p, where in the case of the usual Hamilton–Jacobi nonlinearity, p = 2. Under certain conditions on the exponent p we prove the short-time existence of weak and strong solutions to this family of equations. We also show the uniqueness of strong solutions. Moreover, we prove the blow-up in finite time of certain solutions to this family of equations when the exponent p>2. Furthermore, we discuss the difference in the formation and structure of the singularity between the viscous and hyper-viscous versions of this type of equation.

We prove that for conformal expanding maps the return time does have constant multifractal spectrum. This is the counterpart of the result by Feng and Wu in the symbolic setting.

The differential Galois approach enables us to study the nonintegrability of Hamiltonian systems in a complex analytical meaning, and a Melnikov type approach enables us to detect the occurrence of chaos and their nonintegrability in a real analytical sense. For a class of two-degree-of-freedom Hamiltonian systems with saddle centres we show that the Galoisian obstructions to integrability and Melnikov criteria for chaos are equivalent. We give an example for a pendulum-oscillator type Hamiltonian to illustrate the theory.

We consider the ordinary differential equation

−ü(t) = f(t,u(t)),

where f(t,s) is bounded in t for fixed s and superlinear in s for s→∞. Using a variational method, we prove the existence of infinitely many oscillating solutions belonging to L^∞().

An upper bound of the life-span of smooth solutions to the complex Ginzburg–Landau equation with periodic boundary condition in one space dimension is given explicitly in terms of an integral mean of the Cauchy data in the case where the interaction is focusing.

The exact solution for the scattering of electromagnetic waves on an infinite number of parallel half-planes was obtained by J F Carlson and A E Heins in 1947 using the Wiener–Hopf method (Carlson J F and Heins A E 1947 Quart. Appl. Math.4 313–79). We analyse their solution in the semiclassical limit of small wavelength and find the asymptotic behaviour of the reflection and transmission coefficients. The results are compared with those obtained within the Kirchhoff approximation.

An existence and uniqueness theorem for Pokrovskii's zero-temperature anisotropic gap equation is proved. Furthermore, it is shown that Pokrovskii's finite-temperature equation is inconsistent with the Bardeen–Cooper–Schrieffer (BCS) theory. A reformulation of the anisotropic gap equation is presented along the line of Pokrovskii and it is shown that the new equation is consistent with the BCS theory for the whole temperature range. As an application, the Markowitz–Kadanoff model for anisotropic superconductivity is considered and a rigorous proof of the half-integer-exponent isotope effect is obtained. Furthermore, a sharp estimate of the gap solution near the transition temperature is established.

The long-time behaviour of the system of degenerate reaction–diffusion equations describing detonation in porous media is considered. An upper bound of the bulk burning rate is found.

We study a dissipative nonlinear equation modelling certain features of the Navier–Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions n⩽4. For dimensions n>4, we present strong numerical evidence supporting the existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding the existence of self-similar singular solutions to a semi-linear heat equation.

Group theoretic means are employed to analyse the Hopf bifurcation on pattern forming systems with the periodicity of the face-centred (FCC) and body-centred (BCC) cubic lattices. We find all -axial subgroups of the normal form symmetry group by first extending the symmetry to a larger group. There are 15 such solutions for the FCC lattice, of which at least 12 can be stable for appropriate parameter values. In addition, a number of subaxial solutions can bifurcate directly from the trivial solution, and quasiperiodic solutions can also exist. We find 33 -axial solutions for the BCC lattice and their stability criteria. We discuss applications of the method of symmetry enlargement to other systems. A model-independent approach is taken throughout, and the results are applicable to a wide variety of pattern forming systems. This work is an extension of that done in Callahan T K (2000 Hopf bifurcations on the FCC lattice Proc. Int. Conf. on Differential Equations (Berlin, 1999) vol 1, ed Fiedler et al (Singapore: World Scientific) pp 154 6; 2003 Hopf bifurcations on cubic lattices Bifurcations, Symmetry and Patterns (Trends in Mathematics) ed J Buescu et al (Basel: Birkhauser) pp 123–7).

We consider a lattice gas interacting via a Kac interaction J_γ(|x−y|) of range γ⁻¹, γ>0, x,y∊^d and under the influence of an external random field given by independent bounded random variables with a translation invariant distribution. We study the evolution of the system through a conservative dynamics, i.e. particles jump to nearest neighbour empty sites, with rates satisfying a detailed balance condition with respect to the equilibrium measure. We prove that rescaling space as γ⁻¹ and time as γ⁻², in the limit γ→0, for dimension d⩾3, the macroscopic density profile ρ satisfies, a.s. with respect to the random field, a nonlinear integral differential equation, with a diffusion matrix determined by the statistical properties of the external random field. The result holds for all values of the density, also in the presence of phase segregation, and the equation is in the form of the flux gradient for the energy functional.

We consider the dynamical system x_ttt = c²−½x²−x_t for the parameter c close to zero. We perform a multiple timescale analysis to provide analytic forms for all bounded solutions of the formal normal form in the phase space, in a neighbourhood of the origin (x,c) = (0, 0). These take the form of Jacobi elliptic functions describing periodic and quasi-periodic solutions, and hyperbolic functions that describe heteroclinic connections. A comparison between these approximate analytical results and numerical simulations of the unperturbed system shows excellent correspondence.

A new method of obtaining lower bounds for the attractor's dimension is suggested which involves analysis of homoclinic bifurcations. The method is applied for obtaining sharp estimates of the attractor's dimension for a class of abstract damped wave equations which are beyond the reach of the classical methods.

We prove the existence of analytic solutions of difference equations of the form y(x+ε) = f(x,y(x)), where x,y are complex variables and ε is a small parameter; these solutions are defined on certain bounded or unbounded x-domains and tend to a limit uniformly as ε tends to 0. We also show that differences of two such solutions are exponentially small. We apply these results to the problem of delayed bifurcation at a point of period doubling for real discrete dynamical systems. In contrast to previous publications, the results obtained in this article are global.

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