Tau-functions and dressing transformations for zero-curvature affine integrable equations

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda- and KdV-type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras g. Their common feature is that they have some special vacuum solutions corresponding to Lax operators lying in some Abelian (up to the central term) subalgebra of g; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of g. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the Abelian and non-Abelian affine Toda theories are discussed in detail. © 1997 American Institute of Physics.

Fourth-order method for solving the Navier-Stokes equations in a constricting channel

A fourth-order numerical method for solving the Navier-Stokes equations in streamfunction/vorticity formulation on a two-dimensional non-uniform orthogonal grid has been tested on the fluid flow in a constricted symmetric channel. The family of grids is generated algebraically using a conformal transformation followed by a non-uniform stretching of the mesh cells in which the shape of the channel boundary can vary from a smooth constriction to one which one possesses a very sharp but smooth corner. The generality of the grids allows the use of long channels upstream and downstream as well as having a refined grid near the sharp corner. Derivatives in the governing equations are replaced by fourth-order central differences and the vorticity is eliminated, either before or after the discretization, to form a wide difference molecule for the streamfunction. Extra boundary conditions, necessary for wide-molecule methods, are supplied by a procedure proposed by Henshaw et al. The ensuing set of non-linear equations is solved using Newton iteration. Results have been obtained for Reynolds numbers up to 250 for three constrictions, the first being smooth, the second having a moderately sharp corner and the third with a very sharp corner. Estimates of the error incurred show that the results are very accurate and substantially better than those of the corresponding second-order method. The observed order of the method has been shown to be close to four, demonstrating that the method is genuinely fourth-order. © 1977 John Wiley & Sons, Ltd.

Some results on stability of retarded functional differential equations using dichotomic map techniques

This paper deals with the study of the stability of nonautonomous retarded functional differential equations using the theory of dichotomic maps. After some preliminaries, we prove the theorems on simple and asymptotic stability. Some examples are given to illustrate the application of the method. Main results about asymptotic stability of the equation x′(t) = -b(t)x(t - r) and of its nonlinear generalization x′(t) = b(t) f (x(t - r)) are established. © 1998 Kluwer Academic Publishers.

Solid mass falling into the reservoirs shallow water equations validity from the engineering point of view

Water waves generated by a solid mass is a complex phenomenon discussed in this paper by numerical and experimental approaches. A model based on shallow water equations with shocks (Saint Venant) has developed. It can reproduce the amplitude and the energy of the wave quite well, but because it consistently generates a hydraulic jump, it is able to reproduce the profile, in the case of high relative thickness of slide, but in the case of small relative thickness it is unable to reproduce the amplitude of the wave. As the momentum conservation is not verified during the phase of wave creation, a second technique based on discharge transfer coefficient α, is introduced at the zone of impact. Numerical tests have been performed and validated this technique from the experimental results of the wave's height obtained in a flume.

Conservation laws for non-potential operators

A study was developed in order to build a function M invariant in time, by means of Hamiltonian's formulation, taking into account the equation associated to the problem, showing that starting from this function the equation of motion of the system with the contour conditions for non-conservative considered problems can be obtained. The Hamiltonian method is extended for these kind of systems in order to validate for non-potential operators through variational approach.

Morse potential energy spectra through the variational method and supersymmetry

The Variational Method is applied within the context of Supersymmetric Quantum Mechanics to provide information about the energy and eigenfunction of the lowest levels of a Hamiltonian. The approach is illustrated by the case of the Morse potential applied to several diatomic molecules and the results are compared with stabilished results. (C) 2000 Elsevier Science B.V.

Induced variational method from supersymmetric quantum mechanics and the screened Coulomb potential

The formalism of supersymmetric quantum mechanics supplies a trial wave function to be used in the variational method. The screened Coulomb potential is analyzed within this approach. Numerical and exact results for energy eigenvalues are compared.

Equivalence of the Duffin-Kemmer-Petiau and Klein-Gordon-Fock equations

A strict proof of the equivalence of the Duffin-Kemmer-Petiau and Klein-Gordon Fock theories is presented for physical S-matrix elements in the case of charged scalar particles minimally interacting with an external or quantized electromagnetic field. The Hamiltonian canonical approach to the Duffin - Kemmer Petiau theory is first developed in both the component and the matrix form. The theory is then quantized through the construction of the generating functional for the Green's functions, and the physical matrix elements of the S-matrix are proved to be relativistic invariants. The equivalence of the two theories is then proved for the matrix elements of the scattered scalar particles using the reduction formulas of Lehmann, Symanzik, and Zimmermann and for the many-photon Green's functions.

Variational studies and replica symmetry breaking in the generalization problem of the binary perceptron

We analyze the average performance of a general class of learning algorithms for the nondeterministic polynomial time complete problem of rule extraction by a binary perceptron. The examples are generated by a rule implemented by a teacher network of similar architecture. A variational approach is used in trying to identify the potential energy that leads to the largest generalization in the thermodynamic limit. We restrict our search to algorithms that always satisfy the binary constraints. A replica symmetric ansatz leads to a learning algorithm which presents a phase transition in violation of an information theoretical bound. Stability analysis shows that this is due to a failure of the replica symmetric ansatz and the first step of replica symmetry breaking (RSB) is studied. The variational method does not determine a unique potential but it allows construction of a class with a unique minimum within each first order valley. Members of this class improve on the performance of Gibbs algorithm but fail to reach the Bayesian limit in the low generalization phase. They even fail to reach the performance of the best binary, an optimal clipping of the barycenter of version space. We find a trade-off between a good low performance and early onset of perfect generalization. Although the RSB may be locally stable we discuss the possibility that it fails to be the correct saddle point globally. ©2000 The American Physical Society.

Lamé differential equations and electrostatics

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A(x)y″ + 2B(x)y′ + C(x)y = 0, where A(x),B(x) and C(x) are polynomials of degree p + 1,p and p - 1, is under discussion. We concentrate on the case when A(x) has only real zeros aj and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients rj in the partial fraction decomposition B(x)/A(x) = ∑j p=0 rj/(x - aj), we allow the presence of both positive and negative coefficients rj. The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges rj at aj is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges. © 2000 American Mathematical Society.

Stability of trapped Bose-Einstein condensates

In three-dimensional trapped Bose-Einstein condensate (BEC), described by the time-dependent Gross-Pitaevskii-Ginzburg equation, we study the effect of initial conditions on stability using a Gaussian variational approach and exact numerical simulations. We also discuss the validity of the criterion for stability suggested by Vakhitov and Kolokolov. The maximum initial chirp (initial focusing defocusing of cloud) that can lead a stable condensate to collapse even before the number of atoms reaches its critical limit is obtained for several specific cases. When we consider two- and three-body nonlinear terms, with negative cubic and positive quintic terms, we have the conditions for the existence of two phases in the condensate. In this case, the magnitude of the oscillations between the two phases are studied considering sufficient large initial chirps. The occurrence of collapse in a BEC with repulsive two-body interaction is also shown to be possible.

Galilean covariance and non-relativistic Bhabha equations

We apply a five-dimensional formulation of Galilean covariance to construct non-relativistic Bhabha first-order wave equations which, depending on the representation, correspond either to the well known Dirac equation (for particles with spin 1/2) or the Duffin-Kemmer-Petiau equation (for spinless and spin 1 particles). Here the irreducible representations belong to the Lie algebra of the 'de Sitter group' in 4 + 1 dimensions, SO(5, 1). Using this approach, the non-relativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.

Vibrational-rotational structure of supersingular plus Coulomb potential A/r4 - Z/r*

The vibrational-rotational states of the supersingular plus Coulomb potential A/r4 - Z/r are variationally constructed using a nonorthogonal basis of atomic hydrogenic eigenfunctions modulated by an exponential factor exp(- α/r), ensuring the correct behavior in the vicinity of the supersingularity. The construction is carried out in two successive stages. The first stage is restricted to trial functions without radial nodes, leading to a variational optimization of the parameters of the basis for each value of the angular momentum. The second stage uses the complete basis to construct linear trial functions and to formulate the variational problem in terms of secular equations, yielding the successive vibrational and rotational states. Numerical results for the corresponding energy levels are presented for different combinations of the intensity parameters of the potential. © 2001 Plenum Publishing Corporation.

Bound constrained smooth optimization for solving variational inequalities and related problems

Variational inequalities and related problems may be solved via smooth bound constrained optimization. A comprehensive discussion of the important features involved with this strategy is presented. Complementarity problems and mathematical programming problems with equilibrium constraints are included in this report. Numerical experiments are commented. Conclusions and directions of future research are indicated.

Solutions for a class of integro-differential equations with time periodic coefficients

In this work, a series solution is found for the integro-differential equation y″ (t) = -(ω2 c + ω2 f sin2 ωpt)y(t) + ωf (sin ωpt) z′ (0) + ω2 fωp sin ωpt ∫t 0 (cos ωps) y(s)ds, which describes the charged particle motion for certain configurations of oscillating magnetic fields. As an interesting feature, the terms of the solution are related to distinct sequences of prime numbers.