Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

Separation of variables in the stationary Schrödinger equation

The problem of separation of variables in the stationary Schrödinger equation is considered for a charge moving in an external electromagnetic field. On the basis of the definition formulated, necessary and sufficient conditions are found for separation of variables in equations of elliptic type to which the stationary Schrödinger equation belongs. Application of general theorems made it possible to enumerate all types of electromagnetic fields and systems of coordinates in which separation of variables in the stationary Schrödinger equation is possible. Systems of ordinary differential equations which the wave function in the separated variables satisfies are written down to explicit form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 45–50, August, 1972.     

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.     

On Reduction of the Euler Equations by Means of Two-Dimensional Algebras

Abstract

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler equations are reduced to systems of partial differential equations in two independent variables which are integrated in quadratures.     

Dressed mode description of optical bistability

We generalize and simplify the definition of mode variables given in Haken's theory of phase transitions in systems far from thermal equilibrium. The Maxwell-Bloch equations for absorptive optical bistability in a ring cavity are rephrased in such a way that the boundary conditions for the field become a simple periodicity condition in space without retardation in time. From this formulation of the Maxwell-Bloch equations we derive the time evolution equations for the mode variables, which describe the dressed mode dynamics. The coefficients of these equations are analytically evaluated in the limit of small transmittivity of the mirrors. Some applications are indicated.     

Separation of variables in the Klein-gordon equations. III

This paper supplements [1] and [2]. All kinds of external electromagnetic fields are found which contain arbitrary functions admitting complete separation of variables in the Klein-Gordon equations by using one first and two second order differential symmetry operators. The curvilinear coordinates in which the variables separate are presented, and equations in the separated variables are written down.     

On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables

Abstract

Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented.     

Reductions and conserved quantities for discrete compound KdV–Burgers equations

We present two methods to reduce the discrete compound KdV–Burgers equation, which are reductions of the independent and dependent variables: the translational invariant method has been applied in order to reduce the independent variables; and a discrete spectral matrix has been introduced to reduce the number of dependent variables. Based on the invariance of a discrete compound KdV--Burgers equation under infinitesimal transformation with respect to its dependent and independent variables, we present the determining equations of transformation Lie groups for the KdV--Burgers equation and use the characteristic equations to obtain new forms of invariants.     

Separation of variables in the Klein — Gordon equation. I

All types of external electromagnetic fields containing arbitrary functions which admit of separation of variables in the Klein-Gordon equation by using three first-order differential symmetry operators, and stationary fields admitting separation of variables by using two first- and one second-order differential operators, are found. The curvilinear coordinates in which the variables are divided are presented and the equations are written down in the separated variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 66–72, November, 1973.     

The symmetries of Dynkin diagrams and the reduction of Toda field equations

The exist generalizations of the Toda lattice equations involving the Cartan matrices constructed from the simple and extended root systems of any simple Lie algebra. Toda's original equations correspond to the large-N limit of SU(N). All these equations are known to constitute the integrability conditions for a certain linear problem and as such to have remarkable properties. The symmetries of the equations are investigated by studying the corresponding Dynkin diagrams which conveniently encode the structure of the equations. Corresponding to each conjugacy class of this symmetry group, “reductions” of the equations may be made whereby identification of symmetrically related variables leads to new, self-consistent equations which are integrable in the same sense as before. The new equations which can be regarded as multicomponent generalizations of the Bullough-Dodd equation are shown to correspond precisely to the generalized Cartan matrices classified in the mathematical literature.     

The Maxwell-Vlasov equations as a continuous hamiltonian system

The well-known Maxwell-Vlasov equations that describe a collisionless plasma are cast into hamiltonian form. The dynamical variables are the physical although noncanonical variables E, B and f. We present a Poisson bracket which acts on these variables and the energy functional to produce the equations of motion.     

Hagen-Poiseuille Flow Solutions in Grad-Type Equations

An analysis of the Hagen-Poiseuille flow for a rarefied gas is presented using Grad’s equations and regularized equations in the 13 moment approximation, which provide a correction for the solution in the hydrodynamic regime. Slip boundary conditions are obtained through a simple wall model in which we take into account diffuse and specular wall-particle interactions. From the solution of the regularized equations, we recover as a special case, the velocity profile in Grad’s approximation and also an equivalent expression in the hydrodynamic regime. In addition, other relevant variables are calculated.     

Symmetry Reduction and Exact Solutions of the Euler–Lagrange–Born–Infeld,Multidimensional Monge–Ampere and Eikonal Equations

Abstract

Using the subgroup structure of the generalized Poincaré group P (1, 4), ansatzes which reduce the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which reduce the considered equations to linear ordinary differential equations. The corresponding symmetry reduction has been done. Using the solutions of the reduced equations, some classes of exact solutions of the investigated equation have been presented.     

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.     

Legendre Transformation for Regularizable Lagrangians in Field Theory

Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.     

Dual-phase-lag transfer equations and entropy behavior in relaxational hydrodynamics

Extended thermodynamics of irreversible processes is developed; based on two postulates by which additional variables of the entropy density are dissipative fluxes and material time derivatives of the ordinary thermodynamic variables. Within these theories a more general approximation of entropy production is obtained. As a consequence of the proposed formalism, the constitutive dual-phase-lag equations, as well as equations of the conventional version of extended irreversible thermodynamics are obtained. The behavior of the entropy during oscillatory approach to equilibrium is considered. The proposed theory leads to a strictly monotonic dependency of the entropy on time.     

Reduction and noncommutative integration of linear differential equations

A new method is proposed for derivation of exactly integrable linear differential equations based on the theory of noncommutative integration. The equations are obtained by reduction from original equations which are integrable in the noncommutative sense, with a large number of independent variables. It is shown that the reduced equations cannot be solved by traditional methods, since they do not possess the required algebraic symmetry.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 55–60, November, 1993.     

Application of a fusion model using collective variables and microscopically related mass parameters

A calculation of nucleus-nucleus collisions is presented, using a model which starts from a TDHF equation and leads to classical equations of motion for a set of four collective variables. Restricting to axial symmetry and assuming the liquid drop mass formula to hold, a differential equation is derived, which describes nuclear deformations and energies and is used to construct a potential energy surface for the collective variables. The nuclear deformations are obtained without the need of shape parameters. The equations of motion for the collective variables are solved numerically.