Table of contents

We study local spatial dynamics of the vorticity directions in the 3D Navier-Stokes model. More precisely, utilizing the spatial analyticity of solutions, we obtain estimates of the 2D Hausdorff measure of the level sets arising in the local alignment.

In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax equations, exhibiting the integrability properties of such manifolds. A number of different field equations for such hypercomplex manifolds are derived, one of which is in Cauchy-Kovaleskaya form which enables a formal general solution to be given. Various other properties of the field equations and their solutions are studied, such as their symmetry properties and the associated hierarchy of conservation laws.

We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections into ^N. We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of to X. The same quantity also bounds the factor by which the Hausdorff dimension of X can decrease under these projections. Such a bound is motivated by our discovery that the Hausdorff dimension of X need not be preserved by typical projections, in contrast to classical results on the preservation of a Hausdorff dimension by projections between finite-dimensional spaces. We give an example for any positive number d of a set X with box-counting and Hausdorff dimension d in the real Hilbert space ² such that for all projections into ^N, no matter how large N is, the Hausdorff dimension of (X) is less than d (and in fact, is less than two, no matter how large d is).

The homoclinic orbits of the integrable nonlinear Schrödinger equation are of interest because of the chaotic behaviour of the equation when it is damped and driven near such an orbit (McLaughlin D W and Overman E A 1995 Whiskered tori for integrable Pde's: chaotic behaviour in near integrable Pde's Surveys in Applied Mathematics vol 1 ed Keller et al (New York: Plenum) ch 2). It is known that exact expressions can be obtained for these homoclinic orbits via Bäcklund transformations using squared eigenfunctions of the associated linear operator (see above reference). In this paper, the stationary equations of the NLS hierarchy are used to separate the temporal and spatial flows into completely integrable finite-dimensional systems which define the saturation of a given linear instability of a plane wave. Near the homoclinic orbits satisfying even boundary conditions, the phase space is a product of two-dimensional phase planes similar to the phase plane of the Duffing oscillator. The near homoclinic orbits of the plane wave are realized numerically by a simple Runge-Kutta scheme that solves the two systems along the characteristic directions of space and time.

We give a definition of information entropy for points in metric spaces. This entropy measures the amount of information needed to specify a point of a metric space. For the definition of entropy we need to introduce an additional structure on the metric space which will be called a computable structure. If information sources are considered as a metric space we prove that our entropy has the same value as Shannon's one for almost all points of the space. If general metric spaces are considered there is a relation between entropy and dimension.

We apply variational methods to converse KAM theory. These are useful for symplectomorphisms in the annulus that satisfy weaker hypotheses than those usually required. For instance, we do not need the existence of a global Lagrangian generating function. We obtain the variational principles from the primitive function of our symplectomorphism. They are introduced not only for the orbits of a symplectomorphism, but also for the so-called invariant Lagrangian graphs (ILG). Among the non-degenerate ILG we focus on the minimizing ones. Applications are also described for a broad class of examples.

We prove that the Hausdorff dimension of a conformal repeller is stable under random perturbations. Our perturbation model uses the notion of a bundle random dynamical system. The main ingredient of our proof is a version of the Bowen-Ruelle formula for expanding almost conformal bundle random dynamical systems.

One-parameter families of periodic solutions emanating from equilibrium points of a Hamiltonian system are investigated. A class of families that cannot merge on continuation is indicated; as a result, a lower bound for the number of families which are continuable to an arbitrary large norm or period is obtained. Via these findings, some generic multiplicity results for the prescribed period and prescribed energy problems are established. In particular, it is proved that a starshaped energy surface carries at least n distinct periodic solutions, and bilateral bounds for their periods are found.

We consider the stability and instability of an equilibrium point of a Hamiltonian system of two degrees of freedom in certain resonance cases. We also consider the stability or instability of a fixed point of an area-preserving mapping in certain resonance cases. The stability criteria are established by Moser's invariant curve theorem and the instability is established by Chetaev's theorem.

The breathing circle is a two-dimensional generalization of the Fermi accelerator. It is shown that the billiard map associated to this model has invariant curves in phase space, implying that any particle will have bounded gain of energy.

A spatially discrete version of the general kink-bearing nonlinear Klein-Gordon model in (1 + 1) dimensions is constructed which preserves the topological lower bound on kink energy. It is proved that, provided the lattice spacing h is sufficiently small, there exist static kink solutions attaining this lower bound centred anywhere relative to the spatial lattice. Hence there is no Peierls-Nabarro (PN) barrier impeding the propagation of kinks in this discrete system. An upper bound on h is derived and given a physical interpretation in terms of the radiation of the system. The construction, which works most naturally when the nonlinear Klein-Gordon model has a squared polynomial interaction potential, is applied to a recently proposed continuum model of polymer twistons. Numerical simulations are presented which demonstrate that kink pinning is eliminated, and radiative kink deceleration is greatly reduced in comparison with the conventional discrete system. So even on a very coarse lattice, kinks behave much as they do in the continuum. It is argued, therefore, that the construction provides a natural means of numerically simulating kink dynamics in nonlinear Klein-Gordon models of this type. The construction is compared with the inverse method of Flach, Zolotaryuk and Kladko. Using the latter method, alternative spatial discretizations of the twiston and sine-Gordon models are obtained which are also free of the PN barrier.

In a previous paper (Almeida L 1996 Topological sectors for Ginzburg-Landau energies Rev. Mat. Iberoamericana to appear (preliminary version in author's thesis, ENS Cachan, January 1996)) we studied the components of level sets of Ginzburg-Landau energy functionals on multiply connected domains, and showed that they can be (partially) classified by an extended notion of topological degree. We used this to show the existence of stable states and mountain-pass solutions of Ginzburg-Landau equations. In this work, partly inspired by the techniques we developed with Bethuel (Almeida L and Bethuel F 1998 Topological methods for the Ginzburg-Landau equation J. Math. Pures Appl.77 1-49), we first improve the classification into topological sectors of our earlier mentioned paper, and then obtain quite precise estimates on the threshold transition energies between different sectors. These enable us to, in the setting of the simple models considered, obtain the existence of states whose condensed wavefunction has a non-vanishing topological degree and which are separated from the ground state by very high-energy barriers - this phenomenon can be linked to the great stability of permanent currents in superconducting rings.

We study the global existence and asymptotic behaviour in time of solutions of the Cauchy problem for the relativistic nonlinear Schrödinger equation in one space dimension where the real-valued functions f and g are such that |f^(j)(z)|Cz^1+-j, j = 0,1,2,3, for z+0, where >0, and gC⁵([0,)). Equation (A) models the self-channelling of a high-power, ultra-short laser in matter if for all z 0. When = 0,f = 0 equation (A) also has some applications in condensed matter theory, plasma physics, Heisenberg ferromagnets and fluid mechanics. We prove that if the norm of the initial data is sufficiently small, where then the solution of the Cauchy problem (A) exists globally in time and satisfies the sharp L time-decay estimate ||u(t)||_L C(1+|t|)^-1/2. Furthermore, we prove the existence of the modified scattering states and the nonexistence of the usual scattering states by introducing a certain phase function when 0. On the other hand, the existence of the usual scattering states when = 0 follows easily from our results. AMS classification scheme number: 35Q55

An exact N loop soliton solution to the Vakhnenko equation (VE) is found, where N2 is an arbitrary positive integer. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an exact explicit N-soliton solution by using Hirota's method. The exact N loop soliton solution to the VE is then found in implicit form by means of a transformation back to the original independent variables. The shifts that occur when the solitons interact are found. The general results and details of the interaction between solitons are illustrated for the case N = 3.

This paper deals with a family of nonholomorphic maps of , perturbations of zz². We prove that the Hausdorff dimension of the limit set for iterations of some maps of the family is greater than one.