Non-linear spinor equations with localized solutions

A method is outlined for finding non-linear spinor field equations which have stationary, localized solutions (solitons or droplets) of definite spin and parity.This work has been sponsored in part by the National Science Foundation, GF 42060     

Non-linear equations and covariance

A discussion about the connection between non-linear wave equations covariant under the action of a Lie group on one hand, and the theory of non-linear representations of the covariance group developed elsewhere [1] on the other hand, is presented here.Applications to a class of relativistic non-linear wave-equations are then suggested.     

Some properties of stochastic equations for coupled chemical reactions far from equilibrium

Solutions of master equations for coupled chemical reactions far from equilibrium with one varying molecule species are studied and used for getting information about nonlinear Fokker-Planck equations and slow time-dependent processes such as extinction to an absorbing state and transition between several steady states. The Fokker-Planck equation solution is compared to that of the master equation in a relative sense and it is shown that they agree quite well in some important situations but that in general the cases can deviate considerably, when, e.g., accounting for the mutual importance of two probability maxima.     

Bistable kinetics of simple reactions on solid surfaces: lateral interactions, chemical waves, and the equistability criterion

In heterogeneous reactions, the rate constants of desorption, diffusion and elementary reaction steps are usually strongly dependent on reactant coverages due to adsorbate-adsorbate lateral interactions. We analyze the effect of this factor on the bistable regime of the reaction kinetics. As an example, we consider CO oxidation on Pt(111). The equistability lines in the bistable region for this reaction are calculated by analyzing propagation of chemical waves and taking into account the coverage dependence of the CO diffusion coefficient. The results of simulations are compared with the available experimental data. We show that it is possible to obtain the relationship between various kinetic parameters, for example, between CO and oxygen sticking probabilities, by studying special features of the bistable kinetics.     

Transition factor method for discrete master equations; and application to chemical reactions

We introduce transition factors and derive equations for them which are equivalent to the originalN-dimensional discrete master equation. After transition to continuous variables we obtain nonlocal partial differential equations for these transition factors which are slowly varying variables. Finally we consider a chemical reaction system. Using this method the corresponding master equation is exactly solvable in a very simple manner.     

行星运动物理问题的简单处理

从角动量和能量守恒定律出发,建立行星椭圆运动的轨道方程,导出行星运动时的各个物理量表达式,从而给出了一种以初等数学为基础的处理行星运动的简洁方法。     

Relativistic bound-state equations for fermions with instantaneous interactions

In thispaper three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary spin, parity, and charge parity are derived. For the special case of spinless states bound by interactions with a Coulomb-type potential the properties of the ground-state solutions of the three equations are investigated both analytically and numerically. The coupling-constant spectrum turns out to depend strongly on the spinor structure of the fermion interaction. If the latter is chosen such that the nonrelativistic limits of the equations coincide, an analogous spectrum is found for the instantaneous Bethe-Salpeter and the quasi-potential equations, whereas the two-particle Dirac equation yields qualitatively different results.     

On the physical interpretation of some simple soliton solutions of Einstein's equations

The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.This research was supported through a fellowship from the Consejo Nacional de Investigaciones Cientificas y Technicas de la Republica Argentina.     

Flow equations for electron-phonon interactions

A recently proposed method of continuous unitary transformations is used to decouple the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlich transformations. The two approaches are compared. Our result will turn out to be less singular than Fröhlich's. Furthermore the interaction between electrons belonging to a Cooper pair will always be attractive in our approach. Even in the case where Fröhlich's transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter.     

Non-linear superposition in non-linear evolution equations

Special non-linear evolution equations are discussed and classified in integrable and non-integrable partial differential equations. Important methods of solution are sketched. An interesting task is the calculation of the so-called superposition functions permitting to construct from a few known solutions whole chains of further solutions. For integrable partial differential equations they are computed with the help of BÄcklund transformations providing a rather general superposition rule for special types of solutions. At present however a consistent treatment of non-linear evolution equations is only possible in 1 + 1 space and time dimensions. The extension to higher space dimensions brings many problems. An insight into the results and problems in the one-dimensional case is given together with an outlook indicating the conceptual difficulties for a treatment in higher space dimensions.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

On numerical solution of the shallow water equations with chemical reactions on icosahedral geodesic grid

The shallow water equations coupled to the set of reaction–advection–diffusion equations are discretized on a geodesic icosahedral mesh using the finite volume technique. The method of solution of this coupled system is based on the principle of semi-discretization. The algorithm is mass conserving and stable for advection with the Courant numbers up to 2.7. The important part of the methodology is the optimization of the node positions in the icosahedral grid. It is shown that a slight adjustment of the mesh is instrumental in improving the accuracy of the numerical approximation. The convergence of the approximation of the differential operators is evaluated and compared to the data published in the literature. Numerical tests performed with the shallow water solver include two advection experiments, steady and unsteady zonal balanced flow, mountain flow, and the Rossby wave. The mountain flow and the Rossby wave cases are used to test the transport properties of the method in the case of both passive and reactive scalar fields. The investigation of essential numerical characteristics of the method is concluded by the simulation of an unstable zonal jet. The numerical simulation is performed using the set of shallow water equations without dissipation as well as with the viscosity term added to the momentum equation. Results show that the behavior of the model is consistent with both the literature published on the subject and the general empirical evidence.     

Renormalization equations for models with renormalizable and nonrenormalizable interactions

We study models including renormalizable and nonrenormalizable polynomial interactions. We derive the partial differential equations, which are relevant for the variation of parameters of the model. A supersymmetric model is considered as example.     

A class of collision processes with memory and application to simple chemical reactions in a solvent

We apply the general formalism of a class of non-Markovian processes which we have studied elsewhere to three simple models of chemical reactions: dissociation, isomerization, and diffusion in a double-well potential. Our method leads to explicitly solvable models and numerical computations. The results are in agreement with numerical simulation and stochastic dynamics studies of other authors.     

Non-linear wave and Schrödinger equations

We investigate stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under non-linear perturbations. We show in the case of the wave equations that such solutions are unstable for generic perturbations. For the Schrödinger equations periodic solutions are stable while the quasiperiodic ones are not. We extend these results to periodic solutions of non-linear equations.Partially supported by NSERC under Grant NA7901     

Generalized Hedin's equations for quantum many-body systems with spin-dependent interactions

Hedin's equations for the electron self-energy and the vertex have originally been derived for a many-electron system with Coulomb interaction. In recent years, it has been increasingly recognized that spin interactions can play a major role in determining physical properties of systems such as nanoscale magnets or of interfaces and surfaces. We derive a generalized set of Hedin's equations for quantum many-body systems containing spin interactions, e.g., spin-orbit and spin-spin interactions. The corresponding spin-dependent GW approximation is constructed.     

Integral equations for four identical particles with separable two-particle interactions

The problem of four identical spinless particles with separable two-particle interactions is considered. The integral equations of motion for four particles in the bound states and in the continuum states with two non-ihteracting subsystems in the initial channel are formulated. The angular-momentum reduction of the obtained equations is given.     

Generalized Ginzburg-Landau equations for phase transition-like phenomena in lasers,nonlinear optics,hydrodynamics and chemical reactions

The recently found close analogies between the continuous mode laser, the Bénard instability, and chemical instabilities with respect to their phase transition-like behaviour are shown to have a common root. We start from equations of motion containing fluctuations. We first assume external parameters permitting only stable solutions and linearize the equations, which define a set of modes. When the external parameters are changed the modes getting unstable are taken as order parameters. Since their relaxation time tends to infinity the damped modes can be eliminated adiabatically leaving us with a set of nonlinear coupled order parameter equations resembling the time dependent Ginzburg-Landau equations with fluctuating forces. In two and three dimensions additional terms occur which allow for e.g. hexagonal spatial structures. We also treat the hard mode instability and obtain the stationary distribution function as solution of the Fokker-Planck equation. Our procedure has immediate applications to the Taylor instability, to various chemical reaction models, to the parametric oscillator in nonlinear optics and to some biological models. Furthermore, it allows us to treat analytically the onset of laser pulses, higher instabilities in the Bénard and Taylor problems and chemical oscillations including fluctuations.