Time Asymptotics for Solutions of Vlasov–Poisson Equation in a Circle

We prove that there exists a class of solutions of the nonlinear Vlasov–Poisson equation (VPE) on a circle that converges weakly, as t , to a stationary homogeneous solution of VPE. This behavior is called, in the linear case, Landau damping. The result is obtained by constructing a suitable scattering problem for the solutions of the Vlasov–Poisson problem. A consequence of this result is that a class of stationary solutions of the Vlasov–Poisson equation is unstable in a weak topology.     

Geometric Integrability of the Camassa–Holm Equation

It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a Miura transform and a modified Camassa–Holm equation are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.     

The Inverse Spectral Method for Colliding Gravitational Waves

The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 x < y 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possesses a Lax pair formulation. Using the simultaneous spectral analysis of this Lax pair we study the above Goursat problem as well as its linearized version. It is shown that the linear problem reduces to a scalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problem reduces to a 2 × 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.     

The Fokker–Planck–Kolmogorov Equation with Nonlocal Nonlinearity in the Semiclassical Approximation

A scheme for constructing quasi-classical concentrated solutions of the Fokker–Planck–Kolmogorov equation with local nonlinearity is presented on the basis of the complex WKB-Maslov method. Formal, asymptotic in a series expansion parameter D, D 0 solutions of the Cauchy problem for this equation are constructed with a power accuracy O(D ^3/2). A set of the Hamilton–Ehrenfest equations (a set of equations for average and centered moments) derived in this work is of considerable importance in construction of these solutions. An approximate Green's function is constructed and a nonlinear principle of superposition is formulated in the class of semiclassical concentrated solutions of the Fokker–Planck–Kolmogorov equations.     

Numerically stable solution of coupled channel equations: The local reflection matrix

The second order linear Schrödinger equation is transformed to a first order nonlinear differential equation for a quantityp=(iq ^–1 –)/(iq ^–1 +). In a coupled channel problem all quantities occurring in this equation includingq (the WKB wave number) are matrices andp may be calledlocal reflection matrix. This quantity is closely related to the logarithmic derivative of the Schrödinger function but has no singularities in the classically allowed region. In the asymptotic region where the potential is constant the local reflection matrix approaches the physical reflection matrix. In a pure reflection problem (with an infinite potential on one side) this is the fullS-matrix, in a transmission problem (with an activation barrier of finite height) unitarity of theS-matrix can be used to determine most quantities of physical interest fromp. While standard logarithmic derivative methods can become instable for transmission problems the solution with the local reflection matrix is completely stable both for reflection and transmission problem.     

On the problem of coastal refraction in an impedance waveguide

The problem of wave propagation in an irregular plane impedance waveguide is considered. This problem is similar to the classical coastal refraction problem with arbitrary geometry of the coastline. The two-dimensional integral equation is converted asymptotically in the large parameter kr1 to a one-dimensional equation, which contains, besides the conventional terms, a curvilinear integral taken along the interface (coastline). A convenient formula is obtained for the case of a straight coastline. The results of numerical calculations are presented.State University, St. Petersburg. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 11, pp. 1168–1176, November, 1995.     

Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation

We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev space H^k(R²) for k 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.     

Spectral properties of inviscid solutions of the burgers equation with random initial conditions

The Burgers equation with random self-similar initial conditions is investigated numerically in the inviscid limit by a parallel fast Legendre transform algorithm, using Connection Machine CM-200. The use of this equation for solving the problem of nervous impulse propagation through axons is discussed. An attempt is made to simulate recent experiments where the form of the density of propagated nerve impulses, which initially had a power spectrum close to a white noise distribution, appeared similar to the triangular pulses that arise in the inviscid Burgers equation and where the 1/f power law was observed on scales larger than the typical time interval between pulses. It is shown that in the inviscid Burgers equation model the power spectra for different types of initial conditions in the developed Burgers turbulence regime (i.e., at a sufficiently large time) consists of two parts with a rather sharp transition between them: The spectrum virtually coincides with the initial spectra for low wavenumbers, and the 1/f² law holds for high wavenumbers. There is no interval with an intermediate power law dependence such as 1/f. It is inferred that the true 1/f spectrum of nerve impulses propagating through axons cannot be explained in terms of the Burgers equation model and that other mechanisms must be taken into account.The Stockholm University, Sweden, and the Institute for Continuous Media Mechanics, Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, Nos. 3–4, pp. 225–231, March–April, 1995.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited, II

It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called compatible if they satisfy the Morette–Van Hove formula and anti-compatible if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged.     

Equation of state of condensed media

A new choice is proposed for the generating functional which is used to obtain an integral equation for the radial distribution function that is valid in the domain of dense fluids. The corresponding equation of state for giving the intramolecular potential in the form (r)r ^–s agrees in the case of dense fluids with the known Tait empirical equation of state. The Tait equation is extended to the high-pressure case. Processing the experimental data for water exhibits good agreement between the equation of state obtained and experiment in the pressure range from 10⁵–10⁹ Pa.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 43–47, December, 1981.     

Isotropization of Bianchi Models and a New FRW Solution in Brans–Dicke Theory

Using scaled variables we are able to integrate an equation valid for isotropic and anisotropic Bianchi type I, V, IX models in Brans–Dicke (BD) theory. We analyze known and new solutions for these models in relation with the possibility that anisotropic models asymptotically isotropize, and/or possess inflationary properties. In particular, a new solution of curved (k 0) Friedmann–Robertson–Walker (FRW) cosmologies in Brans–Dicke theory is analyzed.     

TheCP N−1 1/N-action by inverse scattering transformation in angular momentum

The effective action which generates 1/N expansion of theCP _N–1 model in two dimensions is studied here by inverse-problem methods. The action contains a functional determinant, in which auxiliary scalar and vector fields are assumed to have a spherical symmetry. This leads to the introduction, as an associated linear problem, of a radial Schrödinger equation with two potentialsv and , and a potential-dependent centrifugal term {(–r)²/r ²–1/4r ²}. The full inverse scattering formalism is developed here for this diffusion problem. It is formulated in terms of two-component Jost solutions, and leads to a matricial Gel'fand-Levitan-Marchenko equation. The scattering data associated to the potentials by this IST are then used to obtain a closed local form for the whole effective action. This is indeed possible for theCP _N–1 model, owing to the classical integrability. Moreover it is found that no spherically symmetric instanton exists in this case. However the absence of supplementary informations on the 1/N series, due to the non-integrability at quantum level, does not allow safe quantitative conclusions on the general behaviour of the 1/N series at large orders.Laboratoire associé au CNRS UA 280     

Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field

A rigorous path integral representation of the solution of the Cauchy problem for the pure-imaginary-time Schrödinger equation _t (t, x)=–[H–mc ²](t,x) is established.H is the quantum Hamiltonian associated, via the Weyl correspondence, with the classical Hamiltonian [(cp–eA(x))²+m ² c ⁴]^1/2+e(x) of a relativistic spinless particle in an electromagnetic field. The problem is connected with a time homogeneous Lévy process.     

Distribution of a nonstationary electron beam in a dense gas

The problem of the temporal and spatial dependences of the parameters of the action of a modulated fast-electron beam on a dense gas is posed on the basis of the transport equation. The problem is simplified by making it nondimensional and by transforming to the Fokker-Planck approximation. A Green's function formalism is developed for this problem and is used to express the solution of the general nonstationary problem in the form of a convolution of a nonstationary boundary flow with a stationary Green's function. The use of the derived equation is illustrated using as an example the solution of a problem with the simplest stationary Green's function corresponding to the straight-ahead approximation. This approximation is used to consider a general relativistic case with model scattering cross sections. The methods and results of a numerical computer solution of the nonstationary problem of electron retardation in the upper layer of the atmosphere are surveyed.Translated from Trudy Ordena Lenina Fizicheskogo Instituta im. P. N. Lebedeva AN SSSR, Vol. 145, pp. 172–188, 1984.     

Inverse problem of mechanics in momentum space

The problem is solved of reproducing a spherically symmetric potential by means of the velocity vector hodograph in a classical mechanics approximation. A definite transcendental equation is obtained for the radial velocity dependence. Considered as an illustration is the problem of the Maxwell fish eye. A relativistic generalization is given. The algorithms constructed can be used to retrieve potentials possessing internal symmetries in the classical or quantum cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 46–50, January, 1985.In conclusion, I am grateful to Prof. Yu. N. Demkov for active and stimulating interest in the research.     

On Vlasov–Manev Equations, II: Local Existence and Uniqueness

We prove that the initial value problem associated with the Vlasov–Manev system (a Vlasov equation in which a correction of type /r ² is added to the Newtonian or Coulomb potential) has a local in time classical and unique solution for sufficiently regular initial data.     

Generalized Brownian motion and elasticity

We introduce a family of stochastic processes which are a natural extension of Brownian motion to a tensor form. This allows us to solve a Dirichlet problem of linear elasticity obeying Lamé's equation, [1–(d– 1)]²V(x)+ [·V(x)]=0.     

Separation of Variables for Bi-Hamiltonian Systems

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called N manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.     

Coordinate-dependent 3+1 formulation of the general relativity equations for a perfect fluid

This paper defines mass, momentum, and energy densities for a perfect fluid, and derives a coordinate-dependent 3+1 decomposition of the equation of motion in terms of a scalar potential c ² [(–g _g44) 1/2 –1] and a vector potentialA _i cg _4i /(–g ₄₄)^1/2. The momentum equation has the form of the Euler equation except there is an additional force proportional to the vector potential and the rate of change of kinetic energy per unit volume. The momentum and energy equations are integrated to obtain the equations previously derived for a particle. The momentum equation is solved for the total acceleration of a fluid element. The equations are exact and do not depend on the choice of coordinate system.