Stochastic differential equations

In chapter I stochastic differential equations are defined and classified, and their occurrence in physics is reviewed. In chapter II it is shown for linear equation show a differential equation for the averaged solution is obtained by expanding in ατ_c, where α measures the size of the fluctuations and τ_c their autocorrelation time. This result is the underlying reason for the existence of “renormalized transport coefficients”. In chapter III the same treatment is adapted to nonlinear equations. In chapter IV an alternative treatment is described, applicable only in a special case, but not confined to small ατ_c.The emphasis is on physical usefulness rather than mathematical rigor. Throughout the text applications are given at the points where they appeared to serve best as illustrations of the method. The list of references is not complete, but hopefully representative of the literature.     

A remark on abstract operator completely integrable equations

We look at the isospectral deformation equation as compatibility conditions on abstract operators rationally depending on λ. Several concrete examples of abstract completely integrable operator equations in two and three dimensions are presented (non-linear Schrödinger, Korteweg-de Vries, Kadomtzev-Petviashvili, Benjamin-Ono).     

Singular fractional evolution differential equations

We give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.     

Integration of the thirring model equations

It is demonstrated that an exact solution for the Thirring massless model in the canonical quantization scheme follows from its Heisenberg equations in the method of dynamic mapping and the presence of unitary nonequivalent representations of the Dirac massless field.     

Integrable equations and differential geometry

It is shown that the equations which are integrable by the inverse scattering transform method and this method itself admit a natural interpretation in terms of vector and principal zero-curvature fibre bundles with certain structure groups G.     

Symmetry of differential equations. II

The theory of algebraic properties of an equation begun in the first part is continued. The algebra of a linear equation is considered and the possibilities of algebraic classification of the solutions of a system are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 64–70, June, 1977.     

Numerical integration of stochastic differential equations

The stability of incompressible turbulent fluids with respect to weak mean flow perturbations is discussed. It is shown that for a statistically homogeneous, isotropic, and stationary model such perturbations will decay. This is in marked contrast to the compressible case.     

Topological classification of impulsive differential equations

The present paper is concerned with the topological classification of impulsive differential equations. Under the assumption that the linear part of the right-hand side of the equation considered has an exponential dichotomy and the nonlinear perturbation is small enough, it is proved that for the underlying equations there existN + 1 types topologically different from one another.     

On systems of periodic differential equations

Restrictions on the characteristic exponents of periodic differential equations are derived from certain symmetry properties. In one case these restrictions amount to a stability theorem. In other cases they reduce the number of independent exponents.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in österreich, Projekt Nr. 3225.     

Numerical simulations of stochastic differential equations

A simple, very accurate algorithm for numerical simulation of stochastic differential equations is described. Its relationship to colored noise is elucidated and exhibited by explicit results. The especially delicate problem of mean first passage times is highlighted and highly accurate agreement between the numerical simulations and analytic results are shown.     

Symmetry of differential equations. I

The theory of the group properties of differential equations is generalized, and this makes it possible to obtain new results based on symmetry and the separation of variables in the basic equations of theoretical physics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 57–64, June, 1977.     

Spectral expansion for a nonselfadjoint periodic differential operator

In the paper, we construct the spectral expansion for the differential operator generated in L ₂(−∞, ∞) by an ordinary differential expression of arbitrary order with periodic complex-valued coefficients Lebesgue integrable on bounded intervals. Dedicated to the memory of B. M. Levitan     

General Nth-order differential spectral problem: General structure of the integrable equations,nonuniqueness of recursion operator and gauge invariance

The general scalar Gelfand-Dikij-Zakharov-Shabat spectral problem of arbitrary order is considered within the framework of the AKNS method. The general form of the integrable equations is found. Uncertainties which appear in the construction of recursion operator and transformation properties of the integrable equations under the gauge transformations are considered. The manifestly gauge-invariant formulation of the integrable equations is given. It is shown that the intergrable equations under consideration are Hamiltonian ones with respect to the infinite family of Hamiltonian structures.     

Explicit unitary schemes to solve quantum operator equations of motion

To solve operator equations of motion in quantum mechanics and in a quantum scalar field theory, we propose an explicit unitary scheme with arbitrary high order of accuracy.     

The method of finite differences for some operator field equations

We propose a new approach to the formulation of some operator field equations. In Section 1 we study the consistency of some particular difference equations and the convergence of the exact solutions. In Section 2 we apply these results to the quantum mechanical system in one dimension and obtain explicit and general expressions for the transfer operator and for the corresponding matrix elements.