Table of contents

We consider the uniqueness problem of a negative eigenvalue in the spectrum of small fluctuations about a bounce solution in a multidimensional case. Our approach is based on the concept of conjugate points from Morse theory and is a natural generalization of the nodal theorem approach usually used in the one-dimensional case. We show that the bounce solution has exactly one conjugate point at τ = 0 with multiplicity one.

The technique of damage spreading is used to study the phase diagram of the easy axis anisotropic Heisenberg antiferromagnet on two geometrically frustrated lattices. The triangular and kagome systems are built up from triangular units that either share edges or corners, respectively. The triangular lattice undergoes two sequential Kosterlitz–Thouless transitions while the kagome lattice undergoes a glassy transition. In both cases, the phase boundaries obtained using damage spreading are in good agreement with those obtained from equilibrium Monte Carlo simulations.

We investigate the structure of topological defects in the ground states of spinor Bose–Einstein condensates with spin F = 1 or F = 2. The type and number of defects are determined by calculating the first and second homotopy groups of the order-parameter space. The order-parameter space is identified with a set of degenerate ground state spinors. Because the structure of the ground state depends on whether or not there is an external magnetic field applied to the system, defects are sensitive to the magnetic field. We study both cases and find that the defects in zero and non-zero field are different.

We analyse the dynamics and the geometric properties of the potential energy surfaces (PES) of the k-trigonometric model (kTM), defined by a fully connected k-body interaction. This model has no thermodynamic transition for k = 1, a second-order one for k = 2, and a first-order one for k > 2. In this paper we (i) show that the single-particle dynamics can be traced back to an effective dynamical system (with only one degree of freedom), (ii) compute the diffusion constant analytically, (iii) determine analytically several properties of the self-correlation functions apart from the relaxation times which we calculate numerically, (iv) relate the collective correlation functions to those of the effective degree of freedom using an exact Dyson-like equation, (v) using two analytical methods, calculate the saddles of the PES that are visited by the system evolving at fixed temperature. On the one hand we minimize |∇V|², as usually done in the numerical study of supercooled liquids and, on the other hand, we compute the saddles with minimum distance (in configuration space) from initial equilibrium configurations. We find the same result from the two calculations and we speculate that the coincidence might go beyond the specific model investigated here.

The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. From the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. In contrast, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in R² (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.

A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang–Mills equations and the Einstein gravitational field equations are considered.

We consider formal integrals of motion in 2D Hamiltonian dynamical systems, calculated with the normal form method of Giorgilli (1979 Comput. Phys. Commun.16 331). Three different non-integrable and one integrable systems are considered. The time variation DI of the formal integral I is found as a function of the order of truncation N of the integral series. An optimal order of truncation is found from the minima of the variations DI. The level lines of the integral I, representing theoretical invariant curves on a Poincaré surface of section, are compared with the real invariant curves. When chaos is limited, excellent agreement is found between the theoretical and the real invariant curves, if the order of truncation is close to the optimal order. The agreement is poor (a) far from the optimal order and (b) when chaos is pronounced. The optimal order, calculated as a function of the distance R from the origin, decreases when R increases. The decrease is rather smooth in the 1:1 resonance, but it has abrupt steps in the case of a higher order (4:3) resonance. In the case of an integrable Hamiltonian, a formal integral I_F is found which is a function of the exact integral I and of the Hamiltonian, given as power series of the canonical variables. The series converges only within a domain of convergence. The radius of convergence along a particular direction is calculated with the d'Alembert and Cauchy methods. The theoretical invariant curves agree with the real invariant curves only within the domain of convergence of I_F. In the case of non-integrable Hamiltonians, we calculate 'pseudo-radii of convergence' that tend to zero as the order of truncation N increases.

The analytic properties of the lattice Green functions

and

are investigated, where w₁, w₂ are complex variables and α₁, α₂ are real parameters in the interval (0, ∞). In particular, simple and direct methods are developed which enable one to evaluate G₁(α₁, w₁) and G₂(α₂, w₂) in terms of products of two complete elliptic integrals of the first kind. Kampé de Fériet series are also used to derive new transformation formulae which give connections between G₁(α₁, w₁) and G₂(α₂, w₂).

The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the N-fold application of the transformation is also established, and these formalisms are applied for a general quadratic system (a generalized harmonic oscillator) and a quadratic system with an inverse-square interaction up to N = 2. Among the new features found, it is shown, for the general quadratic system, that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution, which is related to the existence of coherent states in the system.

We study equations with infinitely many derivatives. Equations of this type form a new class of equations in mathematical physics. These equations originally appeared in p-adic and later in fermionic string theories and their investigation is of much interest in mathematical physics and applications, in particular in cosmology. Differential equations with an infinite number of derivatives can be written as nonlinear integral equations. We perform a numerical investigation of the solutions of these equations. It is established that these equations have two different regimes of solutions: interpolating and periodic. The critical value of the parameter q separating these regimes is found to be q²_cr ≈ 1.37. The convergence of the iterative procedure for these equations is proven.

Dyson–Schwinger equations (DSEs) for propagators are solved for the scalar Φ³ theory and massive Wick–Cutkosky model. With the help of integral representation, the results are obtained directly in Minkowski space in and beyond bare vertex approximation. Various renormalization schemes are employed which differ by the finite strength field renormalization function Z. The S-matrix is puzzled from the Green's function and the effect of truncation of the DSEs is studied. Independent of the approximation, the numerical solution breaks down for a certain critical value of the coupling constant, for which the on-shell renormalized propagator starts to develop the unphysical singularity at very high space-like square of momenta.

Various aspects of the collective behaviour of non-equilibrium nonideal plasmas are studied. The relaxation of kinetic energy to the equilibrium state is simulated by the molecular dynamics (MD) method for two-component non-degenerate strongly non-equilibrium plasmas. The initial non-exponential stage, its duration and the subsequent exponential stage of the relaxation process are studied for a wide range of ion charge, nonideality parameter and ion mass. A simulation model of the nonideal plasma excited by an electron beam is proposed. An approach is developed to calculate the dynamic structure factor in non-stationary conditions. Instability increment is obtained from MD simulations.