Table of contents

Coupled nonlinear Schrödinger equations, linked by cross modulation terms, arise in both nonlinear optics and in Rossby waves in the atmosphere and ocean. Numerically, Akhmediev and Ankiewicz and Haelterman and Sheppard discovered a class of soiltary waves which are composed of a tall, narrow sech-shaped soliton in one mode, bound to a pair of short, wide sech-shaped peaks in the other mode or polarization. Through the method of matched asymptotic expansions, we derive analytical approximations to these solitary waves, and to their periodic generalization, which have been hitherto accessible only through numerical computation.

Gallavotti proposed an equivalence principle in hydrodynamics, which states that forced-damped fluids can be equally well represented by means of the Navier-Stokes equations and by means of time-reversible dynamical systems called GNS. In the GNS systems, the usual viscosity is replaced by a state-dependent dissipation term which fixes one global quantity. The principle states that the mean values of properly chosen observables are the same for both representations of the fluid. In the same paper, the chaotic hypothesis of Gallavotti and Cohen is applied to hydrodynamics, leading to the conjecture that entropy fluctuations in the GNS system verify a relation first observed in non-equilibrium molecular dynamics. We tested these ideas in the case of two-dimensional fluids. We examined the fluctuations of global quadratic quantities in the statistically stationary state of (a) the Navier-Stokes equations and (b) the GNS equations. Our results are consistent with the validity of the fluctuation relation, and of the equivalence principle, indicating possible extensions thereof. Moreover, in these results the difference between the Gallavotti-Cohen fluctuation theorem and the Evans-Searles identity is evident.

AMS classification scheme numbers: 82C05, 76F20

We analyse the structure of minimal-energy solutions of the baby Skyrme model for any topological charge n; the baby multi-skyrmions. Unlike in the (3+1)D nuclear Skyrme model, a potential term must be present in the (2+1)D baby Skyrme model to ensure stability. The form of this potential term has a crucial effect on the existence and structure of baby multi-skyrmions. The simplest holomorphic baby Skyrme model has no known stable minimal-energy solution for n greater than one. The other baby Skyrme model studied in the literature possesses non-radially-symmetric minimal-energy configurations that look like `skyrmion lattices' formed by skyrmions with n = 2. We discuss a baby Skyrme model with a potential that has two vacua. Surprisingly, the minimal-energy solution for every n is radially symmetric and the energy grows linearly for large n. Further, these multi-skyrmions are tighter bound, have less energy and the same large r behaviour than in the model with one vaccum. We rely on numerical studies and approximations to test and verify this observation.

Constantin, Foais and Gibbon proved that the laser equations (Lorenz PDE) define a dynamical system in L² with a C attractor. We extend this theorem to show that the attractor is contained in every Gevrey class, G^s, for 1<s<. This demonstrates a remarkable smoothing mechanism for this hyperbolic system. We consider the consequences of this theorem for finite-dimensionality of the dynamics.

We study piecewise monotone and piecewise continuous maps f from a rooted oriented tree to itself, with weight functions either piecewise constant or of bounded variation. We define kneading coordinates for such tree maps. We show that the Milnor-Thurston relation holds between the weighted reduced zeta function and the weighted kneading determinant of f. This generalizes a result known for piecewise monotone interval maps.

We use the Lorenz system, the Rikitake model and the nonlinear Schrödinger equation to demonstrate that for completely integrable systems, there exist what we call regular mirror systems near movable singularities. The method for finding the mirror systems is very similar to the original Weiss et al's (1983 J. Math. Phys.24 522-6) version of the Painlevé test. It tests the complete integrability and gives a systematic and conceptual proof that the formal Laurent series generated by the Painlevé test are convergent.

Let F be a closed surface. Let V be a smooth vector field on F inducing a flow _t. A theorem due to Poincaré says that if _t is area preserving, then almost every point of F is recurrent under _t.

In this paper we examine the limit sets of orbits, and prove in particular that if V has hyperbolic singularities, and _t has no closed orbits and no saddle connections (or more generally, the graph of saddle connections does not separate the surface), then _t has an orbit which is dense in the entire surface.

We give a classification of all bounded solutions of the equation

u´´´´ + pu´´ + F´(u) = 0,            - < t < ,

in which F is a general quartic polynomial and p is restricted to various subsets of (-,0]. These results are obtained by combining an a priori estimate with geometric arguments in the (u,u´´)-plane.

We introduce a multifractal formalism for potentials defined on shift systems. We prove that the multifractal spectra are a Legendre transform of thermodynamic functions involving the potentials studied. We obtain the fractal distribution of pointwise dimension for g-measures. Such measures are equilibrium states of potentials not necessarily Hölder continuous and generalize Gibbs measures. In connection with phase transition, we also give examples of potentials with a non-unique equilibrium state and non-analytic multifractal spectra.

We consider the nonlinear Schrödinger equation

i u_t = u_xx - mu - f(|u|²)u

on a finite x-interval with Dirichlet boundary conditions. Assuming that f is real analytic with f(0) = 0 and f´(0)0, we show that the equilibrium solution u0 enjoys a certain kind of Nekhoroshev stability. If most of the energy is located in finitely many modes and sufficiently small, then the amplitudes of these modes are almost constant over a time interval,which is exponentially long in the inverse of the total energy.

This result is due to Bambusi, but the proof given here is conceptually and technically simpler. It may also apply to a larger class of nonlinear partial differential equations.

This paper is the first in a series to address questions of qualitative behaviour, stability and rigorous passage to a continuum limit for solitary waves in one-dimensional non-integrable lattices with the Hamiltonian

with a generic nearest-neighbour potential V. Here we establish that for speeds close to sonic, unique single-pulse waves exist and the profiles are governed by a continuum limit valid on all length scales, not just the scales suggested by formal asymptotic analysis. More precisely, if the deviation of the speed c from the speed of sound c_s = (V´´(0))^1/2 is c_s²/24 then as 0 the renormalized displacement profile (1/²)r_c(/) of the unique single-pulse wave with speed c, q_j+1(t)-q_j(t) = r_c(j-ct), is shown to converge uniformly to the soliton solution of a KdV equation containing derivatives of the potential as coefficients, -r_x+r_xxx+12(V´´´(0)/V´´(0)) rr_x = 0. Proofs involve (a) a new and natural framework for passing to a continuum limit in which the above KdV travelling-wave equation emerges as a fixed point of a renormalization process, (b) careful singular perturbation analysis of lattice Fourier multipliers and (c) a new Harnack inequality for nonlinear differential-difference equations.

Recent work on fourth- and fifth-order differential equations of Painlevé type has shown that equations thought to be hierarchical equivalents of the classical Painlevé transcendents have hyperelliptic asymptotics. We show how such asymptotics may be derived via an associated linear problem.

We show that in the sense understood by Robinson (Robinson J C 1998 Nonlinearity11 529-45) `all possible chaotic dynamics can be approximated in ' by a dynamical system dx/dt = k for a certain k <sub>+</sub>. This implies that this type of `approximation' does not give us important information about the system.

In this paper, discrete analogues of Euler-Poincaré and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L:TG that are G-invariant. These discrete equations provide `reduced' numerical algorithms which manifestly preserve the symplectic structure. The manifold G × G is used as an approximation of TG, and a discrete Langragian :G × G is constructed in such a way that the G-invariance property is preserved. Reduction by G results in a new `variational' principle for the reduced Lagrangian :G , and provides the discrete Euler-Poincaré (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed by Marsden et al (Marsden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys.199 351-395) and Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 223-246) which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G = SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian are equivalent to the Moser-Veselov scheme for the generalized rigid body.

We study the dynamical zeta function Z_D(s) related to the periodic trajectories of the billiard flow for several disjoint strictly convex bodies in ³. We show that the analytic properties of Z_D(s) close to the line of absolute convergence Re s = s₀ are similar to the behaviour close to the line Re s = 1 of the inverse Q(s) = 1/R(s) of the classical Riemann zeta function R(s).

In this paper discrete dynamical systems exhibiting `complicated behaviour' are investigated. We present a computer-assisted method to prove that the given system admits the shift map as a subsystem. The method is applied to the Hénon map with the classical parameter values.

We consider an autonomous Hamiltonian system of fourth-order that has a centre with non-semi-simple imaginary eigenvalues and such that some coefficient of the corresponding normal form at the centre is (strictly) negative.

Firstly, we prove the existence of two-dimensional stable and unstable manifolds to the centre, made of orbits converging polynomially to the equilibrium. Then we show that the existence of a homoclinic orbit that is the transverse intersection of the stable and unstable manifolds, implies the existence of an infinite number of `multibump' homoclinic solutions. In particular the topological entropy of the system is positive.

Our approach relies partially on the calculus of variations.

The corrigendum for this article appears in pdf and ps formats below.