Table of contents

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally independent constants of the motion. Among them, two different subsets of N integrals in involution (including the Hamiltonian) can always be explicitly identified. As particular cases, we recover in a straightforward way most of the superintegrability properties of the Smorodinsky–Winternitz and generalized Kepler–Coulomb systems on spaces of constant curvature and we introduce as well new classes of (quasi-maximally) superintegrable potentials on these spaces. Results presented here are a consequence of the Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated with Poincaré and Beltrami coordinates.

We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg–deVries equation and show that it can be consistently extended multi-dimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series of q-discrete Painlevé equations, and Lax pairs for the reduced equations are found. A 2 × 2 Lax pair is given for a with multiple parameters and, also, for versions of and , all for the first time.

The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian -symmetric part, can be transformed into a Hermitian Hamiltonian. This is achieved by introducing a metric which, in general, renders other observables such as the usual momentum or position as non-Hermitian operators. The metric depends on one real parameter, the full range of which is investigated. The explicit functional dependence of the metric and each associated Hamiltonian is given. A specific choice of this parameter determines a specific combination of position and momentum as being an observable; this can be in particular either standard position or momentum, but not both simultaneously. Singularities of the metric are explored and their removability is investigated. The physical significance of these findings is discussed.

Using the inversion relation method, we calculate the ground-state energy for the lattice integrable models, based on baxterization of multicoloured generalization of Temperley–Lieb algebras. The simplest vertex model is analysed and found to have a mass gap.

In this paper, we study the XX model with particular non-diagonal boundaries. Using a fermionization technique for an extended version of this chain we are able to present a new operator which commutes with the extended and also the original Hamiltonian. It is a highly non-trivial conserved quantity in the spin-chain formalism which gives rise to an interesting combinatorial partition function.

A generalized auxiliary equation method is proposed to construct more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including soliton-like solutions, triangular-like solitions, single and combined non-degenerate Jacobi elliptic doubly-like periodic solutions, and Weierstrass elliptic doubly-like periodic solutions.

We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.

The problem of neutral fermions subject to a pseudoscalar potential is investigated. Apart from the solutions for E = ±mc², the problem is mapped into the Sturm–Liouville equation. The case of a singular trigonometric tangent potential (∼tan γx) is exactly solved and the complete set of solutions is discussed in some detail. It is revealed that this intrinsically relativistic and true confining potential is able to localize fermions into a region of space arbitrarily small without the menace of particle–antiparticle production.

This paper aims to study some calculations related to the nonlinear dynamics in Kerr medium. We can equivalently rewrite the nonlinear transformation operators originated from the Hamiltonian H = ℏχN(N − 1) in the normal ordered form which can be easily dealt with. We also give some applications of this approach.

Two mutually noninteracting qubits with identical modest coupling to one and the same reservoir are considered. For a given Hamiltonian and uncorrelated initial state, the mathematically rigorous Davies theory of the weak-coupling and van Hove limit provides a unique Markovian quantum master equation where absolutely none of the usually made additional assumptions and further approximations are introduced. Due to completely positive time evolution also no artificial correlations can arise. Numerical solution of the Markovian master equation shows that the qubits become entangled. In a first short time-interval containing one single maximum of entanglement for reservoir temperature T = 0, different choices of uncorrelated initial states give rise to a remarkable emergence of entanglement of different degree. The quantitative evaluation is analysed in terms of a measure derived from Wootters concurrence. Selected results show that there are even states that acquire the possible maximum. Particularly those states will show a periodic type of 'collapse and revival' behaviour with exponentially decaying envelope at longer times. This has never been reported so far for noninteracting qubits as mediated by simultaneous coupling to an uncontrollable reservoir. Moreover, even selected uncorrelated mixed states of modest degree of mixture may show a similar behaviour, although less pronounced. For T > 0 states with high degree of entanglement at T = 0 in the first time-interval still show a gradually reduced value up to a few tenth of Kelvin but for T ⩾ 33 K no effects can be observed. Finally, initially entangled states will slowly lose their oscillatory degree, again with exponential envelope, as the bipartite system approaches its stationary final state.

An exactly solvable N-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational -space is proposed. It is shown that through the simple substitutions the finite-difference equation for the N-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.

Generation of entanglement between two qubits by scattering an entanglement mediator is discussed. The mediator bounces between the two qubits and exhibits a resonant scattering. It is clarified how the degree of the entanglement is enhanced by the constructive interference of such bouncing processes. Maximally entangled states are available via adjusting the incident momentum of the mediator or the distance between the two qubits, but their fine tunings are not necessarily required to gain highly entangled states and a robust generation of entanglement is possible.

A new type of surface electromagnetic waves localized near the interface of the halves of the same transparent uniaxial magnetic gyrotropic medium is theoretically predicted. The gyration vectors of the halves are perpendicular to the interface and are oppositely directed to each other. Existence of such waves is due to both anisotropy of the medium and non-coincidence of the gyration vector directions on different sides of the interface. Distribution of intensity and variation of polarization of the surface polaritons as a function of the distance from the interface are studied. It is shown that the surface waves under consideration can be excited in uniformly magnetized ferromagnetics with axis of easy magnetization if their frequency does not exceed the magnetic resonance frequency.

We analyse the scattering of a two-dimensional soliton on a potential well. We show that this soliton can pass through the well, bounce back or become trapped and we study the dependence of the critical velocity on the width and the depth of the well. We also present a model based on a pseudo-geodesic approximation to the full system which shows that the vibrational modes of the soliton play a crucial role in the dynamical properties of its interactions with potential wells.

Abel's integral equations arise in many areas of natural science and engineering, particularly in plasma diagnostics. This paper proposes a new and effective approximation of the inversion of Abel transform. This algorithm can be simply implemented by symbolic computation, and moreover an nth-order approximation reduces to the exact solution when it is a polynomial in r² of degree less than or equal to n. Approximate Abel inversion is expressed in terms of integrals of input measurement data; so the suggested approach is stable for experimental data with random noise. An error analysis of the approximation of Abel inversion is given. Finally, several test examples used frequently in plasma diagnostics are given to illustrate the effectiveness and stability of this method.