Related papers: On the derivation of several second order partial …
This is a survey paper based on previous results of the author. In the paper, we define and discuss the generalizations of linear partial differential equations to multidimensional variational problems. We consider two examples of such…
Quantization of gravitational field in the neighbourhood of arbitrary nontrivial solution of Einstein equations is considered, the 2nd order of perturbation theory is calculated. The expression for quantum corrections of the field operator…
We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.
A general covariant extension of Einstein\'{}s field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector $Z_\mu$. Einstein's solutions…
Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein Gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
In the search for exact solutions to Einstein's field equations the main simplification tool is the introduction of spacetime symmetries. Motivated by this fact we develop a method to write the field equations for general matter in a form…
The linear Boltzmann equation for elastic and/or inelastic scattering is applied to derive the distribution function of a spatially homogeneous system of charged particles spreading in a host medium of two-level atoms and subjected to…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
We propose a natural family of higher-order partial differential equations generalizing the second-order Klein-Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing…
Following an approach proposed by Rosser for deriving the transformation equations of volume charge density and current density we derive the transformation equations for the space-time coordinates of the same event, for the mass and the…
Some exact solutions for the Einstein field equations corresponding to inhomogeneous $G_2$ cosmologies with an exponential-potential scalar field which generalize solutions obtained previously are considered. Several particular cases are…
We consider solutions to degenerate anisotropic elliptic equations in order to study their regularity. In particular we establish second-order estimates and enclose regularity results for the stress field. All our results are new even in…
Based on a simple observation that a classical second order differential equation may be decomposed into a set of two first order equations, we introduce a Hamiltonian framework to quantize the damped systems. In particular, we analyze the…
The method of parameter variation for linear differential equations is extended to classes of second order nonlinear differential equations. This allows to reduce the latter to first order differential equations. Known classical equations…
Modifications of General Relativity usually include extra dynamical degrees of freedom, which to date remain undetected. Here we explore the possibility of modifying Einstein's theory by adding solely nondynamical fields. With the minimal…
Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however,…
The Einstein relation, relating the steady state fluctuation properties to the linear response to a perturbation, is considered for steady states of stochastic models with a finite state space. We show how an Einstein relation always holds…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are…