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Although complex scalar functions, varying over the infinite plane r = {x, y}, commonly possess vortices (points of phase singularity), it is possible to devise exceptional functions with no vortices. Similarly, although real scalar functions representing surfaces as small departures from the plane commonly possess umbilic (locally spherical) points, it is possible to devise exceptional functions that do not possess them. Examples of both are given.

We show that spacetime evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a spacetime dependence of a matrix element of fermionic systems related to the orthogonality catastrophe or boundary states with the τ-function of the modified KP-hierarchy. The established relation allows us to apply the apparatus of soliton theory to the study of nonlinear aspects of quantum dynamics. We also describe a bosonization in momentum space—a representation of a fermion operator by a Bose field in the presence of a boundary state.

Discrete-time evolution operators in integrable quantum lattice models are sometimes more fundamental objects than Hamiltonians. In this paper, we study an evolution operator for the one-dimensional integrable q-deformed Bose gas with XXZ-type impurities and find its spectrum. Evolution operators give a new interpretation of known integrable systems, for instance, our system describes apparently a simplest laser with a clear resonance peak in the spectrum.

The idea of duality in one-dimensional nonequilibrium transport is introduced by generalizing the observations by Mukherji and Mishra. A general approach is developed for the classification and characterization of the steady state phase diagrams which are shown to be determined by the nature of the zeros of a set of coarse-grained functions that encode the microscopic dynamics. A new class of nonequilibrium multicritical points has been identified.

We present the results obtained on the magnetization relaxation properties of an XX quantum chain in a transverse magnetic field. We first consider an initial thermal kink-like state where half of the chain is initially thermalized at a very high temperature T_b while the remaining half, called the system, is put at a lower temperature T_s. From this initial state, we derive analytically the Green function associated with the dynamical behaviour of the transverse magnetization. Depending on the strength of the magnetic field and on the temperature of the system, different regimes are obtained for the magnetic relaxation. In particular, with an initial droplet-like state, that is a cold subsystem of the finite size in contact at both ends with an infinite temperature environment, we derive analytically the behaviour of the time-dependent system magnetization.

From a wide class of translationally invariant discrete nonlinear Schrödinger (DNLS) equations, we extract a two-parameter subclass corresponding to Kerr nonlinearity for which any stationary solution can be derived recurrently from a quadratic equation. This subclass, which incorporates the integrable (Ablowitz–Ladik) lattice as a special case, admits exact stationary solutions that are derived in terms of the Jacobi elliptic functions. Exact moving solutions for the discrete equations are also obtained. In the continuum limit, the constructed stationary solutions reduce to the exact moving solutions to the continuum NLS equation with Kerr nonlinearity. Numerical results are also presented for the special case of localized solutions, including sech (pulse, or bright soliton), tanh (kink, or dark soliton) and 1/tanh (called here inverted kink) profiles. For these solutions, we discuss their linearization spectra and their mobility. Particularly, we demonstrate that discrete dark solitons are dynamically stable for a wide range of lattice spacings, contrary to what is the case for their standard DNLS counterparts. Furthermore, the bright and dark solitons in the non-integrable, translationally invariant lattices can propagate at slow speed without any noticeable radiation.

Systems subjected to holonomic constraints follow quite complicated dynamics that could not be described easily with Hamiltonian or Lagrangian dynamics. The influence of holonomic constraints in equations of motions is taken into account by using Lagrange multipliers. Finding the value of the Lagrange multipliers allows us to compute the forces induced by the constraints and therefore, to integrate the equations of motions of the system. Computing analytically the Lagrange multipliers for a constrained system may be a difficult task that depends on the complexity of systems. For complex systems it is, most of the time, impossible to achieve. In computer simulations, some algorithms using iterative procedures estimate numerically Lagrange multipliers or constraint forces by correcting the unconstrained trajectory. In this work, we provide an analytical computation of the Lagrange multipliers for a set of linear holonomic constraints with an arbitrary number of bonds of constant length. In the appendix explicit formulae are shown for Lagrange multipliers for systems having 1, 2, 3, 4 and 5 bonds of constant length, linearly connected.

In this paper, the potential symmetry method is developed to study systems of nonlinear diffusion equations. Potential variables of the systems are introduced through conservation laws; such conservation laws yield equivalent systems-auxiliary systems of PDEs with the given dependent and potential variables as new dependent variables. Lie point symmetries of the auxiliary systems which cannot be projected to the vector fields of the given dependent and independent variables yield potential symmetries of the systems. Classification for systems of nonlinear diffusion equations with two and three components is performed. Symmetry reductions associated with the potential symmetries are presented.

In this paper, a class of differential–difference equations (DDEs) are considered for Lie group analysis. With the help of symbolic computation MATHEMATICA, the continuous Lie point symmetry technique is extended to obtain corresponding infinitesimals. Similarity reductions are derived by solving the characteristic equations. Then some exact solutions are presented by using inverse transformations. In addition, starting from concrete realization of the generalized Virasoro type symmetry algebra [σ(f₁), σ(f₂)] = σ(f'₁f₂ − f₁f'₂), many high-dimensional DDEs can be derived. For example, we give out the (2+1)-dimensional Toda lattice, modified Toda lattice and special Toda lattice in a uniform way.

An ADO (analytical discrete ordinates) solution is used to establish a concise and accurate result for a basic radiative transfer problem in a finite layer described by the grey equation of transfer with general anisotropic scattering. As a specific application, the solution is evaluated for the case of Fresnel boundary conditions to yield numerical results (of a high standard) for several specific cases.

The case of small intercentre distance in the D-dimensional two Coulomb centres problem (Z₁eZ₂)_D (D ⩾ 2) is studied by solving the wave equations using the separations of variables. Asymptotic expansions for the electronic terms and the quantum defect are obtained. Results obtained are compared with previous asymptotic and numerical treatments. Correspondence between energy terms of the three-dimensional system (Z₁eZ₂)₃ and the D-dimensional system (Z₁eZ₂)_D is found.

Gisin and Popescu (Gisin N and Popescu S 1999 Phys. Rev. Lett.83 432) have shown that more information about their direction can be obtained from a pair of anti-parallel spins compared to a pair of parallel spins, where the first member of the pair (which we call the pointer member) can point equally along any direction in the Bloch sphere. They argued that this was due to the difference in dimensionality spanned by these two alphabets of states. Here we consider similar alphabets, but with the first spin restricted to a fixed small circle of the Bloch sphere. In this case, the dimensionality spanned by the anti-parallel versus parallel alphabet is now equal. However, the anti-parallel alphabet is found to still contain more information in general. We generalize this to having N parallel spins and M anti-parallel spins. When the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension, yet in general, more directional information can be found for sets with smaller |N − M| for any fixed total number of spins. We find that the optimal POVMs for extracting directional information in these cases can always be expressed in terms of the Fourier basis. Our results show that dimensionality alone cannot explain the greater information content in anti-parallel combinations of spins compared to parallel combinations. In addition, we describe an LOCC protocol which extracts optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied.

We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level systems described by a pseudo-Hermitian Hamiltonian with real eigenvalues. This allows the generalization of the fermionic coherent states approach to such systems. Pseudo-fermionic coherent states are constructed as eigenstates of two pseudo-fermion annihilation operators. These coherent states form a bi-normal and bi-overcomplete system, and their evolution governed by the pseudo-Hermitian Hamiltonian is temporally stable. In terms of the introduced pseudo-fermion operators, the two-level system Hamiltonian takes a factorized form similar to that of a harmonic oscillator.

The inequality , with L being the grand orbital quantum number, and its conjugate relation for (⟨r²⟩, ⟨p⁻²⟩) are shown to be fulfilled in the D-dimensional central problem. Their use has allowed us to improve the Fisher-information-based uncertainty relation (I_ρI_γ⩾ const) and the Cramer–Rao inequalities (⟨r²⟩I_ρ ⩾ D²; ⟨p²⟩I_γ ⩾ D²). In addition, the kinetic energy and the radial expectation value ⟨r²⟩ are shown to be bounded from below by the Fisher information in position and momentum spaces, denoted by I_ρ and I_γ, respectively.

We present a detailed analysis of the spin models with near-neighbours interactions constructed in our previous paper (Enciso et al 2005 Phys. Lett. B 605 214) by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.

In this paper, we solve the Schrödinger equation using the finite difference time domain (FDTD) method to determine energies and eigenfunctions. In order to apply the FDTD method, the Schrödinger equation is first transformed into a diffusion equation by the imaginary time transformation. The resulting time-domain diffusion equation is then solved numerically by the FDTD method. The theory and an algorithm are provided for the procedure. Numerical results are given for illustrative examples in one, two and three dimensions. It is shown that the FDTD method accurately determines eigenfunctions and energies of these systems.

Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in the previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle–antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory.

The application of a weak integrability concept to the Skyrme and CPⁿ models in four dimensions is investigated. A new integrable subsystem of the Skyrme model, allowing also for non-holomorphic configurations, is derived. This procedure can be applied to the massive Skyrme model as well. Moreover, an example of a family of chiral Lagrangians providing exact, finite energy Skyrme-like solitons with arbitrary value of the topological charge is given. In the case of CPⁿ models a tower of integrable subsystems is obtained. In particular, in (2+1) dimensions a one-to-one correspondence between the standard integrable submodel and the BPS sector is proved. Additionally, it is shown that weak integrable submodels allow also for non-BPS solutions. Geometric as well as algebraic interpretations of the integrability conditions are also given.