Table of contents

We study the initial value problem for the elliptic - hyperbolic Davey - Stewartson system

where , , u is a complex valued function and is a real valued function. When the above system is called a DSI equation in the inverse scattering literature. Our purpose in this paper is to prove global existence of small solutions to this system in the usual weighted Sobolev space , where

Furthermore, we prove time decay estimates of solutions to the system such that

In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve in the plane. It is shown that there do not exist invariant circles near when there is a point on where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not there are examples with orbits that converge to a point of . If the derivative of the radius of curvature is bounded, such orbits cannot exist. There is also a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards. The final section of the paper concerns an impact oscillator whose dynamics are shown to be the same as a dual billiards map.

We study the decay of correlations for certain multi-dimensional noninvertible maps which do not necessarily satisfy Renyi's condition (the bounded distortion property) and do not necessarily satisfy the Markov condition on the definite partitions. Our method is based on the technique of Markov approximations which was developed by Chernov. We relate the slowness of the decay of correlations to the singularity of the invariant density which is caused by the lack of hyperbolicity. We also see that it can be described by the distortion property of the distributions of the invariant densities.

We establish the existence of non-constant closed geodesics on moduli spaces of SU(2) monopoles of arbitrary charge. More generally, we show that the moduli space of strongly centred monopoles of charge k, , contains a totally geodesic submanifold which can be identified with the moduli space of strongly centred 2-monopoles for even k's and with the moduli space of centred 2-monopoles for odd k's. This submanifold consists of monopoles corresponding to k collinear equally spaced particles.

As a method of finding non-regular complexity in unimodal maps, an approach of Fibonacci sequences, i.e. by the operations of concatenation and cyclic shift of symbolic strings, is analysed rigorously. It turns out that all these kneading sequences obtained in this way can be seen as limits of two special kinds of homomorphisms of submonoids applied infinitely many times. A more general theorem of using homomorphisms of submonoid to obtain non-regular kneading sequences is proved. It contains the approach of Fibonacci sequences, the limits of period-doubling and period-n-tupling sequences, the -composition law and the generalized composition law as its special cases. Finally, some open questions are discussed.

In this paper we prove a theorem on the uniqueness of limit cycles surrounding one or more singularities for Liénard equations. By using this theorem we give a positive answer to the conjecture in Dumortier and Rousseau (1990 Nonlinearity 3 1015 - 39), completing the classification of the cubic Liénard equations with linear damping. It also finishes the study of the generic three-parameter unfoldings of the nilpotent focus in the plane.

Breather solutions are time-periodic and space-localized solutions of nonlinear dynamical systems. We show that the concept of anticontinuous limit, which was used before for proving an existence theorem on breathers and multibreather solutions in arrays of coupled nonlinear oscillators, can be used constructively as a high-precision numerical method for finding these solutions. The method is based on the continuation of breather solutions which trivially exist at the anticontinuous limit. It is quite universal and applicable to a wide class of nonlinear models which can be of arbitrary dimension, periodic or random, with or without a driving force plus damping, etc. The main advantage of our method compared with other available methods is that we can distinguish unambiguously the different breather (or multibreather) solutions by their coding sequence. Another advantage is that we can obtain the corresponding solutions whether they are linearly stable or not. These solutions can be calculated in their full domain of existence. We illustrate the techniques with examples of breather calculations in several models. We mostly consider arrays of coupled anharmonic oscillators in one dimension, but we also test the method in two dimensions. Our method allows us to show that the breather solution can be continued while its frequency enters the phonon band (it then superposes to a band edge phonon with a finite amplitude). We also test that our method works when introducing an extra time-periodic driving force plus damping. Our method is applied for the calculation of breathers in so-called Fermi - Pasta - Ulam (FPU) chains, that is, one-dimensional chains of atoms with anharmonic nearest-neighbour coupling without on-site potential. The breather and multibreather solutions are then obtained by continuation from the anticontinuous limit of an extended model containing an extra parameter. Finally, we show that we can also calculate `rotobreathers' in arrays of coupled rotators, which correspond to solutions with one or several rotators rotating while the remaining rotators are only oscillating. The linear stability analysis of the obtained time periodic solutions (Floquet analysis) of all these models will be done in a forthcoming paper.

We introduce a simple geometrical two-dimensional continued fraction algorithm inspired from dynamical renormalization. We prove that the algorithm is weakly convergent, and that the associated transformation admits an ergodic absolutely continuous invariant probability measure. Following Kosygin and Baldwin, its Lyapunov exponents are related to the approximation exponents which measure the diophantine quality of the continued fraction. The Lyapunov exponents for our algorithm, and related ones also introduced in this article, are studied numerically.

We study a system of two conservation laws which is strictly hyperbolic, but not genuinely nonlinear. We solve the Riemann problem for the inviscid system in a unique way and find explicit travelling wave solutions for the viscous system. It is established that the solutions must be searched for in the space of bounded Radon measures.

The global attractor of a dissipative system of ordinary differential equations can be characterized as the set of solutions which permit an extension to a bounded analytic function on a uniform strip in a complex plane. Using this property, we present two methods for constructing sequences of functions, which may be explicitly computed from the system, and from which one can deduce whether a specific point belongs to the attractor or not. Approximation methods obtained in this way are tested on the Lorenz system and compared with those from Foias and Jolly (1995 Nonlinearity 8 295 - 319).

In this paper we generalize the results of an earlier paper to describe a moving breather in a discrete sine - Gordon system. The method uses the calculus of variations and perturbation theory to find the discreteness effects of a breather moving through a nonlinear Klein - Gordon lattice. These results are then used to calculate the energy contained in a discrete breather and this is shown to be less than the corresponding breather in a continuous system.

The second part of the paper considers the variation of kinetic and potential energies with position of breather; these variations are analysed using Peierls - Nabarro-type calculations. The results of this analysis show that small amplitude breathers can move through a lattice almost unhindered by discreteness, thus demonstrating recent results of MacKay and Aubry.

In this paper we discuss the continuity of filled-in Julia sets of functions meromorphic in the complex plane, i.e. rational or transcendental functions, or polynomials. The Main Theorem is: The filled-in Julia set depends continuously on the function provided the function in question has no Baker domain, wandering domain or parabolic cycle (theorem 3.1). The proofs are based on homotopy arguments and do not require any assumption on the number of singular values, actually, they simultaneously work for rational and transcendental functions. By examples we show the Main Theorem to be sharp. In order to illustrate the usage of filled-in Julia sets, applications to (relaxed) Newton's method are described. Using the continuity result a closing lemma for polynomials and entire transcendental functions is proven.

We consider 3-monopoles symmetric under inversion symmetry. We show that the moduli space of these monopoles is an Atiyah - Hitchin submanifold of the 3-monopole moduli space. This allows what is known about 2-monopole dynamics to be translated into results about the dynamics of 3-monopoles. Using a numerical ADHMN construction we compute the monopole energy density at various points on two interesting geodesics. The first is a geodesic over the two-dimensional rounded cone submanifold corresponding to right angle scattering and the second is a closed geodesic for three orbiting monopoles.

A general procedure to construct a generating partition in 2D symplectic maps is introduced. The implementation of the method, specifically discussed with reference to the standard map, can be easily extended to any model where chaos originates from a horseshoe-type mechanism. Symmetries arising from the symplectic structure of the dynamics are exploited to eliminate the remaining ambiguities of the encoding procedure, so that the resulting symbolic dynamics possesses the same symmetry as that of the original model. Moreover, the dividing line of the partition turns out to pass through the stability islands, in such a way as to yield a proper representation of the quasiperiodic dynamics as well as of the chaotic component. As a final confirmation of the correctness of our approach, we construct the associated pruning front and show that it is monotonous.

We present the first purely semiclassical calculation of the resonance spectrum in the diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with . The classical system is unbound and completely chaotic for a scaled energy larger than a critical value . The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron - nucleus distance causes divergences in the periodic orbit formula. Using a regularization technique introduced in (Tanner G and Wintgen D 1995 Phys. Rev. Lett. 75 2928) together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi-Einstein - Brillouin - Keller (QEBK) quantization is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg-like series converging towards the different Landau thresholds.

A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the Cole - Hopf transform which allows for a careful study of the behaviour of the singularities of the solution. The asymptotic behaviour of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinite-dimensional Calogero-type dynamical system. The initial data are provided by high-order asymptotic approximations of the poles at the critical time for the dispersionless solution via the method of steepest descents. The solution is reconstructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solution as the dispersion parameter . A uniform asymptotic expansion as of the dispersive solution is derived in terms of the Pearcey integral in a neighbourhood of the caustic. A continuum limit of the pole expansion and the Calogero system is obtained, yielding a new integral representation of the solution to the inviscid Burgers equation.

Using a scheme given by Lochak, we derive constructively a Nekhorochev-like result of stability in the planetary n-body problem. This allows us to give bounds on the variation of the semi-major axes of the planets over very long times. In this attempt, we first extend the theorems of stability over exponentially long times in the case of nearly integrable degenerate systems. Then, a refined study of the planetary Hamiltonian is needed to carry out the application. More specifically, we give accurate estimates of the complex analyticity widths for the considered Hamiltonian.