Fourier-Galerkin numerical technique for solitary waves of fifth order Korteweg-de Vries equation

The present paper is aimed to display the performance of the Fourier-Galerkin technique developed earlier by the authors for numerical treatment of localized solutions in ²(-∞, ∞) for higher-order equations. The solitary-wave solution of the Kortweg-de Vries fifth order equation is obtained numerically and compared with approximate results of other authors.     

Riemannian Hilbert submanifolds of nonpositive extrinsic curvature

We consider strongly parabolic, Hilbert submanifolds in Riemannian Hilbert manifolds. We prove that their properties are analogous to the known properties in the finite-dimensional case. The main geometric result consists of Theorem 3: a complete, Riemannian, Hilbert submanifold of nonpositive extrinsic curvature and finite codimension in a Hilbert sphere is a great sphere.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 91–100, 1990.The author expresses his thanks to Professor A. A. Borisenko for posing the problem and guiding the work.     

The large-particle method in arbitrary nets and its application to the solution of problems in the aeroelasticity of a parachute

Modifications are introduced into the large-particle method to make it possible to use the method in arbitrary nonrectangular computing nets in the study of the flow of an ideal compressible gas over mobile thin permeable membranes. A problem in the aeroelasticity of a parachute is stated. An example is given of the solution of a problem in the stationary formulation for an axially symmetric parachute in the wake of a cylindrical leading body for supersonic flow.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 16–22     

Fractional Hamiltonian Monodromy

We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy–momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:( − 2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems. Communicated by Eduard Zehnder Submitted: February 25, 2005; Accepted: November 17, 2005     

Investigation and reduction of variance of a weighted estimate in numerical statistical simulation

The efficiency of the “value” and “partial value” modeling related to the construction of a modeling distribution for an auxiliary random variable by multiplying the initial density by a value function is investigated. The value function usually corresponds to a solution of the adjoint equation. Conditions under which the value modeling of the initial distribution reduces the variance compared to the direct simulation are obtained. It is proved that the variance of the weighted estimate is bounded in the case of the partial value modeling. This proposition provides a basis for a method for determining whether or not the variance of the weighted estimate is bounded. This method uses the majorizing adjoint equation. Using a practically important problem in transport theory as an example, the asymptotic optimization of the distribution of the mean free path is presented. The application of the proposed method of the investigation of the variance boundedness for the analysis of the classical exponential transformation method of simulating the mean free path of a particle in the one-dimensional and the spherical variants is discussed.