Table of contents

The formation of rogue waves is studied in the framework of nonlinear hyperbolic systems with an application to nonlinear shallow-water waves. It is shown that the nonlinearity in the random Riemann (travelling) wave, which manifests in the steeping of the face-front of the wave, does not lead to extreme wave formation. At the same time, the strongly nonlinear Riemann wave cannot be described by the Gaussian statistics for all components of the wave field. It is shown that rogue waves can appear in nonlinear hyperbolic systems only in the result of nonlinear wave–wave or/and wave–bottom interaction. Two special cases of wave interaction with a vertical wall (interaction of two Riemann waves propagating in opposite directions) and wave transformation in the basin of variable depth are studied in detail. Open problems of the rogue wave occurrence in nonlinear hyperbolic systems are discussed.

We determine the spectral curve of charge-3 BPS su(2) monopoles with C₃ cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of genus 2 data. The Richelot correspondence, a generalization of the arithmetic mean, is used to solve for this genus 2 curve. Results of other approaches are compared.

We study homoclinic orbits of the Swift–Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and unstable manifolds is exponentially small and the study requires a method capable of detecting phenomena beyond all algebraic orders provided by the normal form theory. We propose an asymptotic expansion for a homoclinic invariant which quantitatively describes the transversality of the invariant manifolds. We perform high-precision numerical experiments to support the validity of the asymptotic expansion and evaluate a Stokes constant numerically using two independent methods.

Within the abstract framework of dynamical system theory we describe a general approach to the transient (or Evans–Searles) and steady state (or Gallavotti–Cohen) fluctuation theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. In addition to its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.

We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius, in terms of Sobolev norms.

We prove a Liouville theorem for a semilinear heat equation with absorption term in one dimension. We then derive from this theorem uniform estimates for quenching solutions of that equation.

This paper deals with the generalized higher-order Kadomtsev–Petviashvili (KP) equation. The strong instability of solitary wave solutions of this equation will be proved.

The problem of invisibility for bodies with a mirror surface is studied in the framework of geometrical optics. A closely related problem concerning the existence of bodies that have zero aerodynamical resistance is also studied here. We construct bodies that are invisible/have zero resistance in two directions, and prove that bodies which are invisible/have zero resistance do not exist in all possible directions of incidence.

The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid–fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.

Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set with a basin of attraction N, we define a stability index σ(x) of a point x ∊ X that characterizes the local extent of the basin. Let B denote a ball of radius about x. If σ(x) > 0, then the measure of B ∖ N relative the measure of the ball is O(^|σ(x)|), while if σ(x) < 0, then the measure of B ∩ N relative the measure of the ball is of this order. We show that this index is constant along trajectories, and we relate this orbit invariant to other notions of stability such as Milnor attraction, essential asymptotic stability and asymptotic stability relative to a positive measure set. We adapt the definition to local basins of attraction (i.e. where N is defined as the set of initial conditions that are in the basin and whose trajectories remain local to X).

This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in and show that the local stability indices (and hence local stability properties) can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle. In doing this, we extend previous results of Krupa and Melbourne (1995 Ergod. Theory Dyn. Syst.15 121–48; 2004 Proc. R. Soc. Edinb. A 134 1177–97) and give criteria for simple heteroclinic cycles in to be Milnor attractors.

In this paper, we show that there are infinitely many -semi-static homoclinic orbits to under the condition that there exists a cohomology c at the boundary of the flat such that h_c(g) > 0 holds for each .

This paper proves that two compact and uniformly disconnected Ahlfors–David regular sets are quasi-Lipschitz equivalent if and only if they have the same Hausdorff dimension.

We consider the two sequences of biorthogonal polynomials and related to the Hermitian two-matrix model with potentials V(x) = x²/2 and W(y) = y⁴/4 + ty². From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_n,n as n → ∞. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t.

We also prove a general result about the interlacing of zeros of biorthogonal polynomials.