Table of contents

Using the inverse scattering transform the authors found one-parameter and the breather-like four-parameter soliton solutions of a perturbed nonlinear Schrodinger equation which describes the pulse propagation in optical fibres in the femtosecond regime.

The canonical treatment of the quantum dynamics of systems with dissipation affected degrees of freedom proceeds within the sum over histories picture of quantum theory. This paper proposes to supplement that approach with a Schrodinger equation approach to the same physics. Starting from a Lagrangian that is of the same form as those obtained by path integration, one can arrive at a quantized Hamiltonian analogue using the standard rules applied to non-dissipative systems. The resulting Hamiltonian is non-diagonal in position representation, so that the corresponding Schrodinger equation is a nonlinear integro-differential equation reminiscent of the Hartree equation. In the case of the normal tunnel junction, with a quasiparticle heat bath, this equation has an exact solution.

A hierarchy of special solutions of the vacuum field equations of general relativity in dimensions n>or=5 is explicitly constructed from the solutions for n=4. The basic feature of these solutions is that they admit a non-trivial 0-analysis. The meaning of this particular constraint and its relation to plane gravitational waves is also discussed.

In the case of quantum field theory at finite temperature the author investigates the thermal Euclidean Green function for massive twisted scalar fields in the spacetime of an infinite straight cosmic string. Then taking the limit, when mass goes to zero, the author obtains the integral form of the mean-square field as a function of temperature.

A deterministic cellular automaton model is presented to simulate the traffic flow in a two-lane roadway. The model is an extended version of the one-dimensional asymmetric exclusion model to take into account the exchange of cars between the first and the second lanes. Using computer simulation, it is shown that the exchange of cars has an important effect on phase transition between the maximal velocity phase and the high-density phase. The phase diagram of the phase transition is found. Also, a simple mean-field theory is presented to analyse the traffic flow of a two-lane roadway.

Starting from the breaking soliton equation, the author obtains a new integrable equation in 2+1 dimensions. Though the equation has no exchange symmetry of the space variables x and y, the model reduces back to the known (1+1)-dimensional sine-Gordon (or Liouville) equation.

Based on a discrete-ordinate approach, it is shown that the parametrized spectral continuum of the neutron transport operator T=(1/ mu )-(c/ mu )¹ integral _-1 Sigma _s( mu ' to mu )d mu ' is pseudo-continuous. The dispersion relation for its pseudo-continuities depends on both c and the discrete mu .

Exact travelling wave solutions in terms of the Weierstrass P-function of the equation eta _t+a₀ eta _x+a₁ eta eta _x+a₂ eta _xxx+b₀ eta _xx+b ₁( eta eta _x)_x+b₂ eta _xxxx=0 are obtained. These solutions may describe particularly both the evolution of a bounded conoidal wave and kink-shaped waves.

The critical behaviour of a dimer-monomer (BC+A to ¹/₂C₂+AB) irreversible catalytic surface reaction is investigated by means of Monte Carlo simulation technique. On a hexagonal lattice (each site having six nearest neighbours), the reaction exhibits a steady reactive state between two critical concentrations y_A¹ and y_A² of A. The critical exponents associated with various order parameters (i.e. coverages) at the second-order phase transition y_A¹ have been obtained. These exponents do not fall in the universality class of Reggeon field theory as is the case of ¹/₂B₂+A to AB, the usual dimer-monomer reaction.

The magnetization and susceptibility of the spin-1 ferromagnetic chain with random anisotropy fields has been investigated by Trotter-Suzuki decomposition and transfer matrix methods. The results indicate a zero temperature magnetization with a susceptibility exponent close to the spin wave value of 2.

It is suggested that it is not necessary to solve the full non-perturbative problem of two-dimensional charge transport in order to obtain global information about the phase and flow diagrams for the quantum Hall system and its relatives. It is argued that the effective quantum field theories encoding the macroscopic properties of these systems are invariant under a symmetry contained in the modular group. New phase and flow diagrams are obtained which exhibit a hierarchy of only odd or only even phases (Hall plateaus). The Kramers-Wannier-like symmetry determines the exact location of all renormalization group fixed points, which appear to be in agreement with available scaling experiments on the quantum Hall system. It also explains the observed 'super-universality', of the delocalization exponent.

The authors study the tunnel effect for an inverted oscillator with complex frequency. The solution of the corresponding non-Hermitian (NH) Schrodinger equation is found by the evolution operator method, based on the SU(1,1) structure of the Hamiltonian and the Wei-Norman theorem (1963). The authors put forward a generalization of dwell time for NH systems built up from their biorthonormal states. The resulting tunnelling time turns out to be complex.

Presents a non-separable one-dimensional two-body problem. It describes the reflection of a two-particle bound state off a discontinuity across which there is a steplike change in the interparticle potential. The model can be used as a primitive description of a pick-up or stripping reaction at an interface and also describes the diffusion of a particle in a plane containing an extended finite depth trap. The reflection coefficient for the discontinuity is obtained in closed form.

Analogue neural networks of associative memory with continuous time dynamics are studied for nonmonotonic transfer functions using the method of self-consistent signal-to-noise analysis. The Hebb learning rule with unbiased random patterns is assumed for the synaptic couplings. A novel phenomenon is found to occur as a result of a phase transition concerning the property of the local field distribution. In retrieval states of the newly found phase which the authors refer to as the super retrieval phase, noise in the local field vanishes and the memory retrieval without errors ensures even for an extensive number of memory patterns stored under the local learning rule. The storage capacity is obtained as a function of the parameter representing the degree of nonmonotonicity of the transfer functions, with the result that an enhancement of the storage capacity can also occur.

The author has suggested a substitution, reducing generalized Thirring equations to a linear Klein-Gordon equation. Using the above substitution one can obtain infinitely many exact solutions of the nonlinear Thirring system without applying the inverse scattering method.

A condition of Yang-Baxterization of reflection equations for R with two distinct eigenvalues is given. Constant solutions to reflection equations for R related to fundamental representations of sl_q(n) and sl_q(1,1) and for R of the eight-vertex model are Yang-Baxterized.

The superposition of bosons and p=2-parafermions has led to the so-called parasupersymmetric quantum mechanics. The authors show that the structures subtending these developments contain a hidden dynamical SU(3)-symmetry effectively associated with the fundamental irreducible representation 3 (unitarily equivalent to the Gell-Mann one). The representation 3* is also visited. The generalization to arbitrary orders p of paraquantization is briefly discussed.

Analytical properties of a new kind of perturbed Korteweg-de Vries equation with a term of the Hilbert transformation with respect to an unknown function are investigated. It is shown that such an equation possesses periodic and solitary wave solutions and those solutions are modulationally unstable.

Mutually avoiding pairs of parallel walks on a number of d-dimensional lattices are mapped onto undirected random walks which return to the origin on a projected (d-1)-dimensional lattice. Generating functions for the number of such pairs of a given length are thus expressed in terms of standard Green's functions. Directed lattices considered include the directed hypercubic and body-centred hypercubic. The generating function for a directed triangular lattice is also obtained. This work is a generalization of known results for the square lattice.

A generalized deformed algebra SU_Phi (2), characterized by a structure function Phi , is obtained. The usual SU(2) and SU_q(2) algebras correspond to specific choices of the structure function Phi . The action of the generators of the algebra on the relevant basis vectors, as well as the eigenvalues of the Casimir operator, are easily obtained. Possible applications in improving phenomenological models are discussed.

The author has investigated the universal amplitude relation of the free energy per correlation volume (f_s⁺-f_s^-) xi ², for two-dimensional systems presenting an asymmetric behaviour in the renormalization group trajectories above (+) and below (-) the fixed point. As an example, this amplitude is exactly computed in a system possessing logarithmic specific heat singularity. The author also discusses an application of the result in the renormalization group crossover of the tricritical Ising model to the Ising fixed point.

Making use of permutation group technique, the eigenvalue problem of Heisenberg spin-¹/₂ chains can be solved. Especially the ground state of the antiferromagnetic chains can be easily written.

By introducing a natural spectral parameter in the quantum Yang-Baxter equation, the authors construct a family of solutions of Zamolodchikov's tetrahedron equation. The general procedure is applied to the universal quantum group R-matrix.

The authors consider a gaseous mixture where the constituting particles interact through binary elastic collisions and removal processes. This system is described by an extended Boltzmann equation for which they derive exact solutions of similarity type, and determine existence conditions. Finally they write explicit solutions for some particular choices of the collision and removal frequencies.

The authors study the kinetics of the single-layer random sequential adsorption of 'hot' dimers on one dimension by a Monte Carlo simulation. 'Hot' dimers are molecules which after adsorption dissociate and each of the remaining 'hot' monomers fly apart up to a maximum distance R from the original adsorption site. The surface coverage, by the adsorbed particles, grows with time and finally reaches a jamming limit when no more adsorption is possible. The jamming coverage ( theta _J) depends on R according to theta _J=1-constant R^-x (R to infinity ,x equivalent to 0.9). The sticking coefficient and the mean free path of the 'hot' species is also studied. Adsorption of dimers having a certain probability to undergo a fight after dissociation is also analysed.

A new type of branched chain processes is introduced, based on the assumption that at the critical state, memory effects are present and thus the probability lambda _q that at the qth generation a new particle is created depends on the generation index q. This dependence may be described in terms of a kind of hierarchical clustering. The main results are the following: (a) for the supracritical regime the probability xi (N) that the offspring of an initial particle is N has an inverse power law behaviour xi (N) approximately N^-(1+H) Xi (ln N) as N to infinity where H>0 is a fractal exponent related to the mean number nu of particles produced per generation and Xi (ln N) is a periodic function of ln N; (b) for the subcritical regime xi (N) has an exponential tail xi (N) approximately exp(-N ln(1/ nu )); (c) for the critical regime the asymptotic behaviour is described by a stretched exponential xi (N) approximately constant N^-1exp (-KN¹(2-H)/) where 1>H>or=0 is another fractal exponent describing the memory effects.

The authors investigate the kinetics of a generalized monomer-monomer surface reaction model in which there is a variable excluded-volume interaction between same species particles (As and Bs). For no interaction, this system reduces to the monomer-monomer model for which there is a first-order transition between A- and B-saturated phases. As a function of the excluded volume interaction, the first-order transition line for weak interaction terminates at a tricritical point where two second-order transitions meet. These transitions, which appear to be in the Reggeon field theory universality class, separate the A-saturated, reactive, and B-saturated phases. Series expansions and various time-dependent Monte Carlo simulations are used to estimate the critical parameters and exponents associated with these transitions.

For pt.I see Phys. Lett. A, vol.171, p.293 (1992). Computer simulations of reaction-limited colloid aggregation are performed to try to obtain the universal behaviour observed by experimentalists. In a previous communication the effects of reducing the sticking probability were analysed. The effects of a reduction in concentration together with a reduction in sticking probability are studied, reaching concentrations as low as 0.001 in volume fraction and sticking probabilities ten times smaller than those used by other authors. The author still observes the universal behaviour only at the beginning of the aggregation process.

The authors have investigated the current distribution and local power dissipation in the two-component deterministic percolation model in which the ratio of conductance h is regarded as a small parameter. The iterative nature of the model enables them to find exact recursion relations for the current distribution. For an extremely small h such that hL^phi <<1, where L is the size of the lattice and phi the crossover exponent, they find that D( alpha ) approximately=exp(-A( alpha -B(log L)²)/(log L)³), where D( alpha ) d alpha is the number of currents with alpha <-log I_i< alpha +d alpha ; I_i is the current in bond i and A,B are constants. The current distribution is well approximated by a Gaussian with the mean varying with the size as (log L)² while the variance as (log L)³. As a result, both the most probable current and the minimum current scale with L as exp(-constant(log L)²). Similar results are obtained for the distribution of local power dissipation.

The author extends the geometrical approach to the perceptron and shows that, given n examples, learning is of maximal difficulty when the number of inputs d is such that n=5d. He then presents a new perceptron algorithm that takes advantage of the peculiarities of the cost function. In his tests it is more than two times faster than the standard algorithm. More importantly it does not have fixed parameters, like the usual learning constant eta , but it adapts them to the cost function. He shows that there exist an optimal choice for beta , the steepness of the transfer function. He presents also a brief systematic study of the parameters eta and beta of the standard perceptron algorithm.

The authors study a variant of the Mandelbrot percolation process which is of current use as a model of aerogels. The model has two parameters: One of them, Q, is the usual multiscale parameter of the Mandelbrot percolation process and the other, p, is a Bernoulli percolation parameter that is reserved for 'the last step of the construction'. They investigate the phase diagram of this model in the (Q,p) plane. There are two phases, a sol phase and a gel phase, classified according to whether a limit of crossing probabilities vanishes or is nonzero. In the sol phase, they define a correlation length via the rate of decay of a rescaled connectivity function. They show that this length scale diverges at the phase boundary. Furthermore, they demonstrate that if the phase boundary is approached with Q fixed and p tending up to its critical value, P_G(Q), then, up to logarithmic corrections, the divergence is at least as fast as mod P_G-p mod ^-2dH/, where d_H=2- mod log Q/log N mod can be identified as the Hausdorff dimension of the background medium.

A river network model with a variable exponent of drainage basin is presented. The river network model is an extended version of the Scheidegger river model to take into account a flow-dependent meandering. The scaling behaviour of river-size distribution, the scaling of drainage basin and the multifractal structure of flow (channel discharge) distribution are investigated by computer simulation. It is shown that the exponent of drainage basin changes continuously from 1.5 (the value of the Scheidegger river) to 1 (the value of a linear river) with increasing exponent gamma of flow-dependent meandering, and the exponent of cumulative river-size distribution changes from 0.33 to 0.0. It is found that the partition function Z(q) identical to Sigma _iI_i^q of flow distribution scales as Z(q) approximately=L^{zeta (q)} where I_i is the flow (channel discharge) of water passing over the site i within the river network and the summation ranges over all sites. The multiscaling exponent zeta (q) of flow distribution changes with the exponent gamma of flow-dependent meandering. The dependence of f- alpha spectra on values of gamma are discussed.

The authors study a piecewise linear version of an activator-inhibitor model with the aim of analysing the effect of albedo boundary conditions on the formation and stability of patterns. They find concentration profiles for both components and analyse the linear stability properties of those profiles. They show that it is possible, under certain conditions, to control the shape and the stability of the patterns. Also, a scaling behaviour on the marginal stability line has been found.

q-fundamental matrices are introduced and studied. Elementary operations on quantum matrices are discussed. The q-generalization of the classical Laplace's theorem is found. An application of the result is given.

Irreducible representations of Hecke algebras of the type A_n-1, H_n(q), are discussed in the nonstandard basis. Some subduction coefficients, i.e. the transformation coefficients between the standard and nonstandard base of H_n(q), are derived by using the linear equation method.

Quantum deformations of the simplest Lie parasuperalgebra Psqm(2) are constructed by taking advantage of the well known representations of su_q(2, C). As a particular result, the authors show that the fundamental Witten superalgebra sqm(2) cannot be deformed.

Differential realizations of the spl(2,1) superalgebra on the spaces of homogeneous and inhomogeneous polynomials and the corresponding boson-fermion realizations are studied. The new indecomposable and irreducible representations of the spl(2,1) superalgebra are given on subspaces and quotient spaces of the universal enveloping algebra of Heisenberg-Weyl superalgebra. All the finite-dimensional irreducible representations of the spl(2,1) superalgebra are naturally obtained as special cases.

Explicit realization of SU_q(2) algebra is found in terms of operators acting on a two-dimensional sphere and depending on ( theta , phi ) variables. The corresponding eigenfunctions ('spherical q-functions') appear to be a q-generalization of ordinary Legendre polynomials. The recurrence relations and explicit expression in terms of Askey-Wilson polynomials are obtained. The weight function of these polynomials is expressed via double-periodic elliptic functions.

Polynomials, reducing the wavefunction of coupled momenta to products of single ones, are introduced and discussed. Composition of N such polynomials serves as a generating function for 3nj-coefficients of vector addition. For SU(1,1) the polynomials are calculated explicitly and a compact integral representation for 9j-coefficients is obtained as an example. The method is rather general and may be applied to other groups and to higher vector addition coefficients.

A new consequent description of the polarization structure of light within the framework of quantum optics is given by using the polarization gauge SU(2) invariance of free electromagnetic fields and a related concept of the polarization (P) spin. Within this approach the quantum light may be thought as a mixture of the usual polarized photons and unpolarized biphotons. New classes of unpolarized light states generated by P-scalar biphotons are examined. Possible applications are briefly discussed.

Unitary representation of U_q(su(1,1)) with q being roots of unity are studied. The authors construct unitary irreducible highest weight modules and find that the representations are discrete series owing to the unitarity. Moreover, it is revealed that each unitary irreducible highest weight module is equivalent to the tensor product of two modules. They show that one of them is just the unitary irreducible highest weight module of su(1,1) and the other is the same as that of U_q(su(2)). It is also shown that the number of the latter modules is finite in their representation.

The Hamiltonian formalism for a two-dimensional system of ODE possessing an invariant of the motion not containing the time explicitly has suggested a method for the search of first integrals. Applied to the quadratic system it leads to the finding of two phase space configurations.

Generalized symmetries with arbitrary functions of time t for the well known 2+1-dimensional integrable model, Kadomtsev-Petviashvili (KP) equation, are found by means of the extended mastersymmetry approach. Then an explicit and simple constructive formula for the symmetries of the KP equation is derived directly from the symmetry definition equation, without using complicated recursion operators. All the known symmetries appear as special cases of those obtained in this paper. The general infinite-dimensional Lie algebra constituted by these symmetries is also given.

The author observes that when the expansion variable in Painleve analysis satisfies a system of Riccati equations, truncation at a level higher than constant level is allowed. This extends the range of exact solutions to nonlinear partial differential equations that one is able to obtain using truncated Painleve expansions.

The author considers a linear spectral problem associated with the Levi hierarchy of equations. A Bargmann system under the so-called Bargmann constraint between the eigenfunctions and potentials is proposed, and is proved to be a completely integrable Hamiltonian system in the Liouville sense. Moreover, by the involutive solution of the compatible systems under the Bargmann constraint he presents the involutive representation of solutions of the Levi hierarchy. Especially, he obtains the involutive representation of the solution of the well known Burgers equation u_t=-u_xx+2uu_x.

A replica solution for the averaged eigenvalue spectrum of a large symmetric N*N matrix with an exponential distribution p(M_ij)=( square root N/M_0ij)exp(- square root NM_ij/M_0ij) (M_0ij=M₀) of the elements is presented. This problem is reduced to the solution for the averaged eigenvalue spectrum of a homogeneous matrix M_0ij=M₀ with an added Gaussian random matrix. The main part of the obtained spectral density is the well known semicircular law rho ( lambda )=(1/2 pi M₀²) square root 4M₀²- lambda ². In contrast to the Gaussian random matrix an additional second spectral region in the vicinity of M₀ square root N is observed. The analytic result is verified by numerically obtained spectra of such matrices.

For pt.I see ibid., vol.25, p.4437 (1992). The estimation method of the particle trajectory from the far electromagnetic fields which are produced by the particle, is presented. Using the Lienard-Wiechert superpotentials, the authors have previously presented the estimation method of the particle trajectory for the case that the motion is periodic and non-relativistic. This paper indicates that the estimation method can be expanded to the relativistic case.

The problem of Aharonov-Bohm (AB) scattering on two parallel flux lines of arbitrary magnitude is solved exactly; the expression for the scattering cross-section in the region of geometric shadow is derived. It is shown that in the particular case of two antiparallel lines of the same magnetic flux, though the return flux does not exist, the AB scattering still exists.

The Dirac equation for a system of fermion and dyon is separated into decoupled ordinary differential equation in the Robertson-Walker metric. Energy levels of bound states for a fermion and a dyon with charge Z_d<Z_d^c and for j>or= mod q mod +1/2 are obtained. If dyons are indeed present in the universe, they have an interesting astronomical observation.

It is shown, in terms of the Nambu-Jona-Lasinio theory and the Coleman-Weinberg mechanism, that supersymmetry breaking can be realized dynamically due to radiative correction even in the Wess-Zumino model. Supersymmetry behaviour at finite temperatures is also investigated and it is shown that the supersymmetry broken dynamically at zero temperature can be restored at finite temperatures.

For original paper see Rattray et al. ibid., vol.24, p.4375 (1991). The calculation of the stationary probability distribution for a particle subject to random noise using path-integral methods has been recently discussed in a paper by Rattray and McKane (1991). The author wishes to point out that this calculation is incorrect, and that the proper analysis avoids some puzzling features of the results presented in the above paper.

Using the Rau, Wong and Sherrington (1992) model of the storage of a pair of correlated patterns by an optimized synaptic neural network, the authors argue that all maximally-stable neural networks will show high selectivity between correlated memorized patterns. The dilute-retrieval phase diagram is calculated specifically for a network with binary-valued synapses, and is shown to be qualitatively identical to that for the spherically-constrained synapses of the original work.