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Number 49, December 2001
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P S Bindu, M Senthilvelan and M Lakshmanan
In this Letter, the integrability aspects of a generalized Fisher-type equation with modified diffusion in (1+1) and (2+1) dimensions are studied by carrying out a singularity structure and symmetry analysis. It is shown that the Painlevé property exists only for a special choice of the parameter (m = 2). A Bäcklund transformation is shown to give rise to the linearizing transformation to the linear heat equation for this case (m = 2). A Lie symmetry analysis also picks out the same case (m = 2) as the only system among this class having a nontrivial infinite-dimensional Lie algebra of symmetries and that the similarity variables and similarity reductions lead in a natural way to the linearizing transformation and physically important classes of solutions (including known ones in the literature), thereby giving a group theoretical understanding of the system. For nonintegrable cases in (2+1) dimensions, associated Lie symmetries and similarity reductions are indicated.
David S Dean and Satya N Majumdar
We consider the Newtonian dynamics of a massive particle in a one-dimensional random potential which is a Brownian motion in space. This is the zero-temperature nondamped Sinai model. As there is no dissipation the particle oscillates between two turning points where its kinetic energy becomes zero. The period of oscillation is a random variable fluctuating from sample to sample of the random potential. We compute the probability distribution of this period exactly and show that it has a power law tail for large period, P(T ) ~ T -5/3, and an essential singularity P(T ) ~ exp (-1/T ) as T →0. Our exact results are confirmed by numerical simulations and also via a simple scaling argument.
P D Jarvis and J D Bashford
A calculational framework is proposed for phylogenetics, using nonlocal quantum field theories in hypercubic geometry. Quadratic terms in the Hamiltonian give the underlying Markov dynamics, while higher degree terms represent branching events. The spatial dimension L is the number of leaves of the evolutionary tree under consideration. Momentum conservation modulo 2×L in L←1 scattering corresponds to tree edge labelling using binary L-vectors. The bilocal quadratic term allows for momentum-dependent rate constants - only the tree or trees compatible with selected nonzero edge rates contribute to the branching probability distribution. Applications to models of evolutionary branching processes are discussed.
R S Kaushal
The solutions of an analogous Schrödinger wave equation for the one-dimensional non-Hermitian Hamiltonian H(x, p) in the complex phase plane characterized by x = x1 + i p2, p = p1 + ix2, are investigated. The quasi-exact solutions thus obtained reveal a lot about the nature of the complex eigenvalue spectrum of the potential concerned. The examples of harmonic, harmonic plus inverse harmonic and Mörse potentials are discussed.
Amnon Neeman and N I Shepherd-Barron
For more than a decade now, the chiral Potts model in statistical mechanics has attracted much attention. A number of people have written quite extensively about it. The solutions give rise to a curve over . Au-Yang and Perk found a large subgroup of the automorphism group of this curve. In this letter, we compute the automorphism group precisely.
C Verhoeven and M Musette
From the Lax pair and the binary Darboux transformation of a coupled Korteweg-de Vries system, we show that its nonlinear superposition formula is identical to that obtained for the Kaup-Kupershmidt equation. Therefore, the N-soliton solution can be associated with a determinant of the Gram type.
Manuel Alonso and Yoshiyuki Endo
The variance of the position distribution for a Brownian particle is derived in the general case where the particle is suspended in a flowing medium and, at the same time, is acted upon by an external field of force. It is shown that, for uniform force and flow fields, the variance is equal to that for a free particle. When the force field is not uniform but depends on spatial location, the variance can be larger or smaller than that for a free particle depending on whether the average motion of the particles takes place toward, respectively, increasing or decreasing absolute values of the field strength. A few examples concerning aerosol particles are discussed, with especial attention paid to the mobility classification of charged aerosols by a non-uniform electric field. As a practical application of these ideas, a new design of particle-size electrostatic classifier differential mobility analyser (DMA) is proposed in which the aerosol particles migrate between the electrodes in a direction opposite to that for a conventional DMA, thereby improving the resolution power of the instrument.
B Bonnier
The jamming coverage θd for the random sequential adsorption of aligned d-dimensional cubes in Rd is studied through a one-gap distribution function. Heuristic arguments generally used to describe the large time kinetics are found to imply that this distribution is rather independent of the space dimension. This is shown to give a quantitative explanation of the so-called Palasti approximation θd≃θ1d.
R Brak and J W Essam
We enumerate sets of n non-intersecting, t-step paths on the directed square lattice which are excluded from the region below the surface y = 0 to which they are initially attached. In particular we obtain a product formula for the number of star configurations in which the paths have arbitrary fixed endpoints. We also consider the 'return' polynomial,
where is the number of n-path configurations of watermelon type having deviation y for which the path closest to the surface returns to the surface m times. The 'marked return' polynomial is defined by
where is the number of marked configurations having at least m returns, just m of which are marked. Both and are expressed in terms of the numbers of paths ignoring returns but introducing a suitably modified endpoint condition. This enables to be written in product form for arbitrary y, but for this can only be done in the case y = 0.
A C C Coolen and J A F Heimel
We solve the dynamics of the on-line minority game (MG), with general types of decision noise, using generating functional techniques a la De Dominicis and the temporal regularization procedure of Bedeaux et al. The result is a macroscopic dynamical theory in the form of closed equations for correlation and response functions defined via an effective continuous-time single-trader process, which are exact in both the ergodic and in the non-ergodic regime of the MG. Our solution also explains why, although one cannot formally truncate the Kramers-Moyal expansion of the process after the Fokker-Planck term, upon doing so one still finds the correct solution, that the previously proposed diffusion matrices for the Fokker-Planck term are incomplete, and how previously proposed approximations of the market volatility can be traced back to ergodicity assumptions.
F M Marchetti and B D Simons
A statistical field theory is developed to explore the density of states and spatial profile of 'tail states' at the edge of the spectral support of a general class of disordered non-Hermitian operators. These states, which are identified with symmetry broken, instanton field configurations of the theory, are closely related to localized sub-gap states recently identified in disordered superconductors. By focusing separately on the problems of a quantum particle propagating in a random imaginary scalar potential, and a random imaginary vector potential, we discuss the methodology of our approach and the universality of the results. Finally, we address some potential physical applications of our findings.
Tatiana G Rappoport and M A Continentino
In this paper we investigate the magnetic properties of heavy fermions in the antiferromagnetic and dense Kondo phases in the framework of the Kondo necklace model. We use a mean-field renormalization group approach to obtain a temperature versus Kondo coupling (T–J) phase diagram for this model in qualitative agreement with Doniach's diagram, proposed on physical grounds. We further analyse the magnetically disordered phase using a two-site approach. We calculate the correlation functions and the magnetic susceptibility that allow us to identify the crossover between the spin-liquid and the local moment regimes, which occurs at a coherence temperature.
J M Speight and P M Sutcliffe
We prove the existence of discrete breathers (time-periodic, spatially localized solutions) in weakly coupled ferromagnetic spin chains with easy-axis anisotropy. Using numerical methods we then investigate the continuation of discrete breather solutions as the intersite coupling is increased. We find a band of frequencies for which the one-site breather continues all the way to the soliton solution in the continuum. There is a second band, which abuts the first, in which the one-site breather does not continue to the soliton solution, but a certain multi-site breather does. This banded structure continues, so that in each band there is a particular multi-site breather which continues to the soliton solution. A detailed analysis is presented, including an exposition of how the bifurcation pattern changes as a band is crossed. The linear stability of breathers is analysed. It is proved that one-site breathers are stable at small coupling, provided a non-resonance condition holds, and an extensive numerical stability analysis of one-site and multisite breathers is performed. The results show alternating bands of stability and instability as the coupling increases.
Xue-Juan Zhang
In this article, interwell stochastic resonance (SR) as well as intrawell SR in an under-damped stochastic system without periodic modulation is presented according to different types of stable equilibrium point on the one-dimensional global attractor. The corresponding mechanism is convincingly explained based on the discussion of the attracting ability of the global attractor.
V G Bagrov, J C A Barata, D M Gitman and W F Wreszinski
The dynamics of two-level systems in time-dependent backgrounds is under consideration. We present some new exact solutions in special backgrounds decaying in time. On the other hand, following ideas of Feynman et al, we discuss in detail the possibility of reducing the quantum dynamics to a classical Hamiltonian system. This, in particular, opens the possibility of directly applying powerful methods of classical mechanics (e.g. KAM methods) to study the quantum system. Following such an approach, we draw conclusions of relevance for 'quantum chaos' when the external background is periodic or quasi-periodic in time.
K J Barnes, J Hamilton-Charlton and T R Lawrence
This paper analyses the geometry of the Lie algebra of SO(6) by making use of its homomorphism with SU(4). We study the vector space of 4×4 traceless, Hermitian matrices from four different viewpoints and examine the connections between them. We review the strata of this space under group transformations using established techniques for su(N) algebras. We focus on orbits of special types of vectors and their interpretation as rotations of SO(6) spinors.
S De Bièvre and J Renaud
In this paper, we construct a causal and conformally covariant massless free quantum field on the (1 + 1)-dimensional Minkowski spacetime. This field is of the Gupta–Bleuler type and does not suffer from infrared divergences. Although negative frequency states are used in its construction, the field has no unusual physical features: the energy–momentum tensor behaves reasonably when applied to physical states and it is also shown that particle detectors moving in the vacuum behave as expected.
M de Montigny, F C Khanna and A E Santana
We use a covariant-like formulation of Galilei invariance in five dimensions to obtain a model for compressible irrotational barotropic fluids with pressure proportional to the square of the mass density. Some solutions for the one-dimensional version of one of these equations are found by using their Lie point symmetries. Other models of fluids are also discussed. Our purpose is to illustrate how the Galilei-covariant formalism can serve as a guide for the construction of non-relativistic wave equations relevant in many-body theories.
Hongyi Fan and Junhua Chen
By virtue of the technique of integration within an ordered product of operators we derive the normal ordering expansion of the Dirac's radial momentum operator. To realize this goal, we also derive some operator identities of the power of radial coordinate operators. They are useful in calculating expectation values in the coherent state.
B G Pusztai and L Fehér
It is well known that a classical dynamical r-matrix can be associated with every finite-dimensional self-dual Lie algebra by the definition R(ω): = f(ad ω), where ω∊ and f is the holomorphic function given by f(z) = (1/2)coth (z/2)-1/z for z∊∖2πi*. We present a new, direct proof of the statement that this canonical r-matrix satisfies the modified classical dynamical Yang-Baxter equation on .
W García Fuertes, M Lorente and A M Perelomov
The quantum Calogero-Sutherland model of An-type (Calogero F 1971 J. Math. Phys. 12 419-36, Sutherland B 1972 Phys. Rev. A 4 2019-21) is completely integrable (Olshanetsky M A and Perelomov A M 1977 Lett. Math. Phys.2 7-13, Olshanetsky M A and Perelomov A M 1978 Funct. Anal. Appl.12 121-8, Olshanetsky M A and Perelomov A M 1983 Phys. Rep.94 313-404). Using this fact, we give an elementary construction of lowering and raising operators for the trigonometric case. This is similar to, but more complicated (due to the fact that the energy spectrum is not equidistant) than the construction for the rational case (Perelomov A M 1976 ITEP Preprint No 27).
Janusz Grabowski and Giuseppe Marmo
Jacobi algebroids, i.e. graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding Lie brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov.
Carlos R Handy, C Trallero-Giner and Arezky H Rodriguez
Moment based methods have produced efficient multiscale quantization algorithms for solving singular perturbation/strong coupling problems. One of these, the eigenvalue moment method (EMM), developed by Handy and Bessis (Handy C R and Bessis D 1985 Phys. Rev. Lett.55 931) and Handy et al (Handy C R, Bessis D, Sigismondi G and Morley T D 1988b Phys. Rev. Lett.60 253), generates converging lower and upper bounds to a specific discrete state energy, once the signature property of the associated wavefunction is known. This method is particularly effective for multi-dimensional, bosonic ground state problems, since the corresponding wavefunction must be of uniform signature, and can be taken to be positive. Despite this, the vast majority of problems studied have been on unbounded domains. The important problem of an electron in an infinite quantum lens potential defines a challenging extension of EMM to systems defined on a compact domain. We investigate this in this paper, and introduce novel modifications to the conventional EMM formalism that facilitate its adaptability to the required boundary conditions.
A V Ivanov
We consider the double mathematical pendulum in the limit when the ratio of pendulum masses is close to zero and the ratio of pendulum lengths is close to infinity. We found that the limit system has a hyperbolic periodic trajectory, whose invariant manifolds intersect transversally and the intersections are exponentially small. In this case we obtain an asymptotic formula of the homoclinic invariant for the limit system.
I K Johnpillai and F M Mahomed
Semi-invariants for the linear parabolic equations with two independent variables (time variable t and space variable x) and one dependent variable u are derived under the transformation of the independent variables, by using the infinitesimal method. We also obtain the joint invariant equation for the above-mentioned equation under equivalence transformation. In fact, we prove a necessary and sufficient condition for a (1 + 1) parabolic equation to be reducible via a local equivalence transformation to the one-dimensional classical heat equation. This result provides practical criteria for reduction. Finally, examples of (1 + 1) Fokker–Planck equations from applications are given to verify the results obtained.
Xin-zhou Li and Xiang-hua Zhai
The Casimir effect giving rise to an attractive force between the configuration boundaries that confine the massless scalar field is rigorously proved for an odd dimensional hypercube with the Dirichlet boundary conditions and different spacetime dimensions D by the Epstein zeta function regularization.
A Matos-Abiague
A new kind of deformed calculus (the D-deformed calculus) that takes place in fractional-dimensional spaces is presented. The D-deformed calculus is shown to be an appropriate tool for treating fractional-dimensional systems in a simple way and quite analogous to their corresponding one-dimensional partners. Two simple systems, the free particle and the harmonic oscillator in fractional-dimensional spaces, are reconsidered in the framework of the D-deformed quantum mechanics. Confined states in a D-deformed quantum well are studied. D-deformed coherent states are also found.
Sergei Nechaev and Raphaël Voituriez
The stable profile of the boundary of a plant's leaf fluctuating in the direction transverse to the leaf's surface is described in the framework of a model called a `surface à godets' (SG). It is shown that the information on the profile is encoded in the Jacobian of a conformal mapping (the coefficient of deformation) corresponding to an isometric embedding of a uniform Cayley tree into the 3D Euclidean space. The geometric characteristics of the leaf's boundary (such as the perimeter and the height) are calculated. In addition, a symbolic language allowing us to investigate the statistical properties of a SG with annealed random defects of the curvature of density q is developed. It is found that, at q = 1, the surface exhibits a phase transition with the critical exponent α = ½ from the exponentially growing to the flat structure.
S A Pol'shin
We construct the systems of generalized coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in elementary functions and are invariant under the SO(3, 2) (discrete spectrum) and SO(4, 1) (continuous spectrum) subgroups of the dynamical symmetry group SO(4, 2) of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which intertwines irreducible representations of the SO(4, 2) group.
J Rasmussen and M A Walton
Information on su(N) tensor product multiplicities is neatly encoded in Berenstein–Zelevinsky triangles. Here we study a generalization of these triangles by allowing negative as well as non-negative integer entries. For a fixed triple product of weights, these generalized Berenstein–Zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so-called virtual triangles. Inequalities satisfied by the coefficients of the virtual triangles describe a polytope. The tensor product multiplicities may be computed as the number of integer points in this convex polytope. As our main result, we present an explicit formula for this discretised volume as a multiple sum. As an application, we also address the problem of determining when a tensor product multiplicity is non-vanishing. The solution is represented by a set of inequalities in the Dynkin labels. We also allude to the question of when a tensor product multiplicity is greater than a given non-negative integer.
Riccardo Rosso and Epifanio G Virga
Lipid bridges are lipid membranes linking two parallel, adhesive walls. For appropriate values of both physical and geometrical parameters, there are two types of such bridges, which look quite different from one another. Here we apply a general condition valid for two-dimensional lipid architectures to show that when the elastic energy density of the lipid membrane is quadratic in the mean curvature, both these bridges are locally stable. Moreover, we give a criterion to decide about their global stability when they happen to coexist at equilibrium.
T Skrypnyk and P Holod
Using a family of special quasigraded Lie algebras on hyperelliptic curves we construct new hierarchies of integrable nonlinear equations admitting zero-curvature representations. We show that in the case of the rational degeneration of the curve they coincide with Heisenberg magnet hierarchies.
Vladimir V Sokolov and Thomas Wolf
Several classes of systems of evolution equations with one or two vector unknowns are considered. We also investigate systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are listed.
Michael Wilkinson
A new type of basis set for quantum mechanical problems is introduced. These basis states are adapted to describing the dynamics of a Hamiltonian Ĥ which is dependent upon a parameter X. A function f(E) is defined which is an analytic function of E, and which is negligibly small when |E|>>δE, where δE is large compared to the typical level separation. The energy-shell basis set consists of states |ξn(X)⟩ which are derived by applying the operator f(Ĥ(X)-Ēn(X)) to elements of a fixed basis set, where Ēn(X) is an analytic approximation to an eigenvalue En(X). The energy-shell basis states are combinations of states close to energy En, but vary more slowly as a function of X than the eigenfunctions |ϕn(X)⟩ of Ĥ(X). This feature gives the energy-shell basis states some advantages in analysing solutions of the time-dependent Schrödinger equation.
H M Yehia
We show that the well-known Routhian procedure of ignoring cyclic coordinates is far more than a tool for obtaining equations of motion of reduced order in a Lagrangian form. A transformation is introduced that involves a number of arbitrary functions or additional parameters in the system while preserving the Routhian equations of motion. Although these parameters invoke new physical effects in the transformed system, the solution of the latter is always obtained in a simple way from that of the original system. In particular, from any integrable system with k cyclic degrees of freedom we obtain a family of systems integrable on a fixed level of the cyclic integrals and physically generalizing that system through the inclusion of k arbitrary functions that depend only on the noncyclic coordinates. In many problems of physical interest, a general integrable case can also be generalized to an integrable case for arbitrary initial conditions. The method is applied to some problems of rigid body dynamics. Four new integrable problems are obtained as generalizations of known cases by including certain additional combinations of gravitational and electromagnetic forces. The new cases are presented in an explicit way that enables direct verification of the constancy of the integrals using the equations of motion. Explicit time solution of the new cases is discussed. Physical interpretation is given for two cases.