Table of contents

We show that a suitable choice of the zero-order problem can lead directly to the emergence of non-Taylor expansions in λ for the ground-state energies of the Hamiltonian family H(λ) = −∇² + r² + λ/r^β in regions β ⩾ 3. The discussion includes the role of dimension and the right order parameter. The effect of choosing a more general potential of the form r^α + λ/r^β is also considered.

In this letter, we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function.

In this letter, we develop a mode-coupling theory for a class of nonlinear Langevin equations with multiplicative noise using a field-theoretic formalism. These equations are simplified models of realistic colloidal suspensions. We prove that the derived equations are consistent with the fluctuation–dissipation theorem. We also discuss the generalization of the result given here to real fluids, and the possible description of supercooled fluids in the ageing regime. We demonstrate that the standard idealized mode-coupling theory is not consistent with the FDT in a strict field-theoretic sense.

In the framework of the nonlinear Λ-model we investigate propagation of a slow-light soliton in atomic vapours and Bose–Einstein condensates. The velocity of the slow-light soliton is controlled by a time-dependent background field created by a controlling laser. For a fairly arbitrary time dependence of the field we find the dynamics of the slow-light soliton inside the medium. We provide an analytical description for the nonlinear dependence of the velocity of the signal on the controlling field. If the background field is turned off at some moment of time, the signal stops. We find the location and shape of the spatially localized memory bit imprinted into the medium. We show that the process of writing optical information can be described in terms of scattering data for the underlying scattering problem.

Recently, Göhmann, Klümper and Seel have derived novel integral formulae for the correlation functions of the spin-1/2 Heisenberg chain at finite temperature. We have found that the high temperature expansion (HTE) technique can be effectively applied to evaluate these integral formulae. Actually, as for the emptiness formation probability P(n) of the isotropic Heisenberg chain, we have found a general formula of the HTE for P(n) with arbitrary up to O((J/T)⁴). If we fix a magnetic field to a certain value, we can calculate the HTE to much higher order. For example, the order up to O((J/T)⁴²) has been achieved in the case of P(3) when h = 0. We have compared these HTE results with the data by quantum Monte Carlo simulations. They exhibit excellent agreement.

We present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential. We consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit. Apart from the general interest in the effect of geometrical confinement, this can be viewed as a two-dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions. We demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half-plane geometry, even when the polymer is constrained to be fixed at both ends on one wall. We are not able to find a closed form solution for the free energy for finite width, at all values of the interaction parameters, but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane. This allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur.

We consider a section of a half-filled chain of free electrons and its entanglement with the rest of the system in the presence of one or two interface defects. We find a logarithmic behaviour of the entanglement entropy with constants depending continuously on the defect strength.

For a continuum model of one-dimensional anharmonic crystal lattices at finite temperatures, which was derived from a statistical-mechanical model proposed recently, we clarify the classification of its differential system. That is, we determine not only the strict hyperbolicity and convexity regions but also the elliptic and parabolic regions in the space of the state. The melting point is found to be on the boundary of the convexity region. Then we derive the Rankine–Hugoniot relations, and we prove that the admissible shocks are always in the stable region of convexity.

Kinetically grown self-avoiding walks on various types of generalized random networks have been studied. Networks with short- and long-tailed degree distributions P(k) were considered (k, degree or connectivity), including scale-free networks with P(k) ∼ k^−γ. The long-range behaviour of self-avoiding walks on random networks is found to be determined by finite-size effects. The mean self-intersection length of non-reversal random walks, ⟨l⟩, scales as a power of the system size N: ⟨l⟩ ∼ N^β, with an exponent β = 0.5 for short-tailed degree distributions and β < 0.5 for scale-free networks with γ < 3. The mean attrition length of kinetic growth walks, ⟨l⟩, scales as ⟨L⟩ ∼ N^α, with an exponent α which depends on the lowest degree in the network. Results of approximate probabilistic calculations are supported by those derived from simulations of various kinds of networks. The efficiency of kinetic growth walks to explore networks is largely reduced by inhomogeneity in the degree distribution, as happens for scale-free networks.

We investigate the solution of the quantal time-dependent linear oscillator from the viewpoint of the Lie point symmetries of its time-dependent Schrödinger equation. Four of the five nongeneric symmetries can be used for the construction of wavefunctions, two as creation symmetries and two as annihilation symmetries. The fifth nongeneric symmetry provides an eigenvalue for the Lie Bracket which maps a solution symmetry to itself. The general treatment indicates that care must be taken with the selection of solutions of the classical equation for the time-dependent linear oscillator and its third-order self-adjoint counterpart to obtain results which are consistent with the standard results for the autonomous harmonic oscillator.

Using the extended homogeneous balance method (EHBM) and variable separation approach (VSA), an exact variable separation excitation of a (2+1)-dimensional system is derived. Based on the derived excitation, a new general type of solitary wave, i.e., semifolded solitary waves and semifoldons (SFSWs), is defined and studied. We investigate the behaviours of the interactions for the semifolded localized structures and find that the interactions possess some novel and interesting features.

On the basis of the Heisenberg equation of motion and Schrödinger's correspondence , we search for so-called invariant eigenoperators for some Hamiltonians describing photonic nonlinear interactions. In this way the energy-level gap of the Hamiltonians can be naturally obtained. The characteristic polynomial theory has been fully employed in our derivation.

It is shown that the self-dual Yang–Mills (SDYM) equations for the ∗-bracket Lie algebra on a heavenly space can be reduced to one equation (the master equation). Two hierarchies of conservation laws for this equation are constructed. Then the twistor transform and a solution to the Riemann–Hilbert problem are given.

We seek exact solutions to the Einstein field equations which arise when two spacetime geometries are conformally related. Whilst this is a simple method to generate new solutions to the field equations, very few such examples have been found in practice. We use the method of Lie analysis of differential equations to obtain new group invariant solutions to conformally related Petrov type D spacetimes. Four cases arise depending on the nature of the Lie symmetry generator. In three cases we are in a position to solve the master field equation in terms of elementary functions. In the fourth case special solutions in terms of Bessel functions are obtained. These solutions contain known models as special cases.

We relate Miura type transformations (MTs) over an evolution system to its zero-curvature representations with values in Lie algebras . We prove that certain homogeneous spaces of produce MTs and show how to distinguish these spaces. For a scalar translation-invariant evolution equation, this allows us to classify all MTs in terms of homogeneous spaces of the Wahlquist–Estabrook algebra of the equation. For other evolution systems this allows us to construct some MTs. As an example, we study MTs over the KdV equation, a fifth-order equation of Harry Dym type, and the coupled KdV–mKdV system of Kersten and Krasilshchik.

The position and momentum operators of the q-oscillator (with the main relation aa⁺ − qa⁺a = 1) are symmetric but not self-adjoint if q > 1. They have one-parameter family of self-adjoint extensions. These extensions are given explicitly. Their spectra and eigenfunctions are derived. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a⁺ and a of the q-oscillator at q > 1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.

The dynamics of a generalized non-degenerate optical parametric down-conversion interaction whose Hamiltonian includes an arbitrary time-dependent driving part and a two-mode coupled part is studied by adopting the Lewis–Riesenfeld invariant theory. The closed formulae for the evolution of the quantum states and the evolution operators of the system are obtained. It is shown that various generalized squeezed states arise naturally in the process, and the two-mode squeezed effect is independent of the driving part. An explicitly analytical solution of the Schrödinger equation is further derived as the classical generalized force acting on each mode and the coupling of the two modes both have harmonic time dependences. This solution is found to be in agreement with previous research in special cases.

The scattering by a three-dimensional separable potential of rank 2 is described analytically. The model potential is chosen so that the Lippmann–Schwinger equation can be exactly solved and all calculations are carried out analytically. The explicit expressions for a number of scattering properties, such as phase shifts, cross sections, T- and S-matrices, resonances and poles of the resolvent are calculated. The second virial coefficient is also evaluated in terms of phase shifts and bound state energies for the particular potential model presented.

We study the Bogoliubov–Dirac–Fock model introduced by Chaix and Iracane (1989 J. Phys. B: At. Mol. Opt. Phys.22 3791–814) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off Λ and the bare fine structure constant α. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off Λ and without any constraint on the external field. We also study the behaviour of the minimizer as Λ goes to infinity and show that the theory is 'nullified' in that limit, as predicted first by Landau: the vacuum totally cancels the external potential. Therefore, the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant α, on a simplified model where the exchange term is neglected.

We obtain an exact solution of the ideal magnetohydrodynamics equations in (3+1) dimensions. The construction of the solution is based on the invariants of the group O(3) of rotations. It is a deep generalization of the known solution, which describes a radial flow with spherical waves. It is shown that in the irreducible solution, the velocity and magnetic field vectors of the particle are coplanar to the radius vector of the particle. The solution is represented as an involutive invariant subsystem of PDEs with two independent variables and finite relation, which contains a functional arbitrariness.

The new singularities in χ⁽³⁾ found by Zenine et al (2004 J. Phys. A: Math. Gen.37 9651) are pinch-type singularities analogous to those found in Feynman graphs in field theory. A general formula for the possible location of all such singularities for arbitrary χ^(N) is given.