Table of contents

The interplay between chaotic tunnelling and dynamical localization in mixed phase space is investigated. Semiclassical analysis using complex classical orbits reveals that tunnelling through torus regions and transport in chaotic regions are not independent processes, rather they are strongly correlated and described by complex orbits with both properties. This predicts a phenomenon analogous to the quantum suppression of classical diffusion: chaotic tunnelling is suppressed as a result of dynamical localization in chaotic regions. This hypothesis is confirmed by numerical experiments where the effect of destructive interference is attenuated.

A classical upper bound for quantum entropy is identified and illustrated, 0 ⩽ S_q ⩽ ln(eσ²/2ℏ), involving the variance σ² in phase space of the classical limit distribution of a given system. A fortiori, this further bounds the corresponding information-theoretical generalizations of the quantum entropy proposed by Rényi.

Fractional differential equations (FDEs) considered so far contain mostly left (or forward) fractional derivatives. In this paper, we present analytical solutions for a class of FDEs which contain both the left and the right (or the forward and the backward) fractional derivatives. The methods presented use properties of fractional integral operators (which, in many cases, lead to Volterra-type integral equations), an operational approach and a successive approximation method to obtain the solutions. The methods are demonstrated using some examples. The FDEs considered may come from fractional variational calculus (FVC) or from other physical principles. In the case of fractional variational problems (FVPs), the transversality conditions are used to identify appropriate boundary conditions and to solve the problems. It is hoped that this study will lead to further investigations in the field and more elegant solutions would be found.

A reduction method of ODEs not possessing Lie point symmetries makes use of the so-called λ-symmetries (Muriel and Romero 2001 IMA J. Appl. Math.66 111). The notion of covering for an ODE is used here to recover λ-symmetries of as nonlocal symmetries. In this framework, by embedding into a suitable system determined by the function λ, any λ-symmetry of can be recovered by a local symmetry of . As a consequence, the reduction method of Muriel and Romero follows from the standard method of reduction by differential invariants applied to .

This paper is devoted to the construction of new integrable quantum-mechanical models based on certain subalgebras of the half-loop algebra of . Various results about these subalgebras are proven by presenting them in the notation of the St Petersburg school. These results are then used to demonstrate the integrability, and find the symmetries, of two types of physical system: twisted Gaudin magnets and Calogero-type models of particles on several half lines meeting at a point.

Introducing p − 1 new parameters into the multilinear relations, we extend the standard unitary parasupersymmetry algebra of order p so that by embedding the quantum solvable models possessing gl(2, c) Lie algebra symmetry into it, the partitions of integer numbers p − 1 and are established. These two partitions are performed by the new parameters and the product of new parameters with their labels, respectively. The former partition is just necessary for the real form h₄; however, both of them are essential for the real forms u(2) and u(1, 1). By occupying these parameters with arbitrary values, the energy spectra are determined by the mean value of proposed parameters for the real form h₄ with their label weight function as well as for the real forms u(2) and u(1, 1) with the weight function of their squared label. So for the given energies, the multilinear behaviour of parasupercharges is not specified uniquely by varying the new parameters continuously.

We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors, and in terms of them we construct discriminants and the determinant as the discriminant of order d, where d is the dimension of the matrix. Analogues of the characteristic polynomials and the Cayley–Hamilton theorem are obtained therefrom for higher rank matrices.

The Boussinesq equation usually arises in a physical problem as a long wave equation. The present work extends the search of periodic wave solutions for it. The Hirota bilinear method and Riemann theta function are employed in the process. We also analyse the asymptotic property of periodic waves in detail. Furthermore, it is of interest to note that well-known soliton solutions can be reduced from the periodic wave solutions.

Asymptotic behaviour of a new class of anomalous diffusion equations for subdiffusive transport defined in terms of generalized distributed fractional-order time derivatives is considered. The effect of slowly varying factors on the scaling function of asymptotic solutions is demonstrated. The origin of slowly varying scaling factors in the CTRW models is discussed.

An exactly solvable model for the particle transport between two reservoirs of non-interacting Bose particles through a microscopic tunnelling junction is studied. The analysis covers the whole range of temperatures and densities of the reservoirs, and the particle and energy flows between the reservoirs are calculated in closed form. In particular, in the case, when Bose–Einstein condensation is present, their dependence on the phase difference of the condensates is established.

We solve the problem of discrete translocation of a polymer through a pore, driven by the irreversible, random sequential adsorption of particles on one side of the pore. Although the kinetics of the wall motion and the deposition are coupled, we find the exact steady-state distribution for the gap between the wall and the nearest deposited particle. This result enables us to construct the mean translocation velocity demonstrating that translocation is faster when the adsorbing particles are smaller. Monte Carlo simulations also show that smaller particles give less dispersion in the ratcheted motion. We also define and compare the relative efficiencies of ratcheting by deposition of particles with different sizes and we describe an associated 'zone-refinement' process.

We consider a new equation recently found by Schäfer and Wayne, hereafter named Schäfer–Wayne short pulse equation (SWSPE), describing the propagation of an ultrashort pulse in nonlinear media. Using some vanishing boundary conditions, we construct and discuss the N-soliton solutions to the previous equation by means of the Wadati–Konno–Ichikawa (WKI) method, which is arguably more direct than the map through the sine-Gordon equation investigated much earlier by Sakovich and Sakovich (2005 J. Phys. Soc. Japan74 239, 2006 J. Phys. A: Math. Gen.39 L361). We particularly focus our attention on the two-soliton solution as an application. As a result, the collision process for two-soliton solutions with 'similar' amplitudes exhibits very different behaviour from the case when the amplitudes are 'dissimilar'.

We present a systematic way to construct dynamical systems with a specific symmetry group G. Each symmetric strange attractor has a unique image attractor that is locally identical to it but different at the global topological level. Image attractors can be lifted to many inequivalent covering attractors. These are distinguished by an index that has related topological, algebraic and group theoretical interpretations. These methods are used to describe dynamical systems with symmetry groups V₄, S₄ and S₆.

Using perturbative methods, we analyse a nonlinear generalization of Schrodinger's equation that had previously been obtained through information-theoretic arguments. We obtain analytical expressions for the leading correction, in terms of the nonlinearity scale, to the energy eigenvalues of the linear Schrodinger equation in the presence of an external potential and observe some generic features. In one space dimension these are (i) for nodeless ground states, the energy shifts are subleading in the nonlinearity parameter compared to the shifts for the excited states; (ii) the shifts for the excited states are due predominantly to contribution from the nodes of the unperturbed wavefunctions, and (iii) the energy shifts for excited states are positive for small values of a regulating parameter and negative at large values, vanishing at a universal critical value that is not manifest in the equation. Some of these features hold true for higher dimensional problems. We also study two exactly solved nonlinear Schrodinger equations so as to contrast our observations. Finally, we comment on the possible significance of our results if the nonlinearity is physically realized.

We analyse the asymptotics of the Wigner 3j-symbol as a matrix element connecting eigenfunctions of a pair of integrable systems, obtained by lifting the problem of the addition of angular momenta into the space of Schwinger's oscillators. A novel element is the appearance of compact Lagrangian manifolds that are not tori, due to the fact that the observables defining the quantum states are noncommuting. These manifolds can be quantized by generalized Bohr–Sommerfeld rules and yield all the correct quantum numbers. The geometry of the classical angular momentum vectors emerges in a clear manner. Efficient methods for computing amplitude determinants in terms of Poisson brackets are developed and illustrated.

The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The suggested modification is based on methods to calculate low-lying eigenpairs of large bounded Hermitian operators or matrices.

This paper presents a constructive proof of complete kinematic state controllability of finite-dimensional open quantum systems whose dynamics are represented by Kraus maps. For any pair of states (pure or mixed) on the Hilbert space of the system, we explicitly show how to construct a Kraus map that transforms one state into another. Moreover, we prove by construction the existence of a Kraus map that transforms all initial states into a predefined target state (such a process may be used, for example, in quantum information dilution). Thus, in sharp contrast to unitary control, Kraus-map dynamics allows for the design of controls which are robust to variations in the initial state of the system. The capabilities of non-unitary control for population transfer between pure states illustrated for an example of a two-level system by constructing a family of non-unitary Kraus maps to transform one pure state into another. The problem of dynamic state controllability of open quantum systems (i.e., controllability of state-to-state transformations, given a set of available dynamical resources such as coherent controls, incoherent interactions with the environment, and measurements) is also discussed.

By recognizing that stress–energy–momentum tensors are fundamentally related to gravitation in spacetime it is argued that the classical electromagnetic properties of a simple polarizable medium may be parameterized in terms of a constitutive tensor whose properties can in principle be determined by experiments in non-inertial (accelerating) frames and in the presence of weak but variable gravitational fields. After establishing some geometric notation, discussion is given to basic concepts of stress, energy and momentum in the vacuum where the useful notion of a drive form is introduced in order to associate the conservation of currents involving the flux of energy, momentum and angular momentum with spacetime isometries. The definition of the stress–energy–momentum tensor is discussed with particular reference to its symmetry based on its role as a source of relativistic gravitation. General constitutive properties of material continua are formulated in terms of spacetime tensors including those that describe magneto-electric phenomena in moving media. This leads to a formulation of a self-adjoint constitutive tensor describing, in general, inhomogeneous, anisotropic, magneto-electric bulk matter in arbitrary motion. The question of an invariant characterization of intrinsically magneto-electric media is explored. An action principle is established to generate the phenomenological Maxwell system and the use of variational derivatives to calculate stress–energy–momentum tensors is discussed in some detail. The relation of this result to tensors proposed by Abraham and others is discussed in the concluding section where the relevance of the whole approach to experiments on matter in non-inertial environments with variable gravitational and electromagnetic fields is stressed.

In this paper two things are done. We first prove that an arbitrary power p of the Schrödinger Lagrangian in arbitrary dimension always enjoys the non-relativistic conformal symmetry. The implementation of this symmetry on the dynamical field involves a phase term as well as a conformal factor that depends on the dimension and on the exponent. This non-relativistic conformal symmetry is shown to have its origin on the conformal isometry of the power p of the Klein–Gordon Lagrangian in one higher dimension which is related to the phase of the complex scalar field.

The temperature inversion symmetry, for a non-interacting supersymmetric ensemble, at finite volume, is studied. It is found that the scaled free energy, f(ξ), is antisymmetric under temperature inversion transformation, i.e. . This occurs for antiperiodic bosons and periodic fermions in the compact dimension. In contrast, for periodic bosons and antiperiodic fermions, .

We discuss how the ordinary renormalization group (RG) equations arise in the context of Wilson's exact renormalization group (ERG) as formulated by Polchinski. We consider the ϕ⁴ theory in four-dimensional Euclidean space as an example and introduce a particular scheme of parameterizing the solutions of the ERG equations. By analysing the scalar composite operators of dimensions 2 and 4, we show that the parameters obey mass-independent RG equations. We conjecture the equivalence of our parameterization scheme with the MS scheme for dimensional regularization.

Studying electromagnetic waves in complex media has been an important research topic due to its useful applications and scientific significance of its physical performance. Dyadic Green's functions (DGFs), as a mathematical kernel or a dielectric medium response, have long been a valuable tool in solving both source-free and source-incorporated electromagnetic boundary value problems for electromagnetic scattering, radiation and propagation phenomena. A complete eigenfunctional expansion of the dyadic Green's functions for an unbounded and a planar, arbitrary multilayered gyrotropic chiral media is formulated in terms of the vector wavefunctions. After a general representation of Green's dyadics is obtained, the scattering coefficients of Green's dyadics are determined from the boundary conditions at each interface and are expressed in a greatly compact form of recurrence matrices. In the formulation of Green's dyadics and their scattering coefficients, three cases are considered, i.e. the current source is immersed in (1) the first, (2) the intermediate, and (3) the last regions, respectively. Although the dyadic Green's functions for an unbounded gyroelectric medium has been reported in the literature, we here present not only unbounded but also multilayered DGFs for the gyrotropic chiral media. The explicit representation of the DGFs after reduction to the gyroelectric or isotropic case agrees well with those existing corresponding results.

In the line above equation (1d), `Maxwell equation' should read `Maxwell equations'. In equations (3a) and (3b), e^iz should read e^ikz. The wave number k appears correctly in subsequent equations.