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Oleg V. Kaptsov

Publications and source records attributed to Oleg V. Kaptsov.

5 recordsLinked to original sources
arXiv2022

Application of invariants of characteristics to construction of solutions without gradient catastrophe

The one-dimensional system of equations of isentropic gas dynamics is considered. First-order invariants of characteristics of this system are classified. Second-order invariants of characteristics are classified for polytropic processes. The infinite sequence of Darboux integrable systems is described. The approach to construction of smooth solutions without gradient catastrophe is proposed. Examples of solutions without gradient catastrophe are presented.

math-ph↗
arXiv2016

Ideals of partial differential equations

We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gröbner bases to clarify crucial notions concerning compatibility such as passivity (involution) and reducibility. One obtains sufficient conditions for a differential system to be passive and prove that such systems generate manifolds in the jet space. Some examples of constructions of passive systems associated with the sinh-Gordon equation are given.

math.DS↗
arXiv2013

Passive collections of differential power series algebra

We consider a differential algebra F of formal power series in infinitely many variables. We define the important notions of a normalized set of generators for an ideal of F and a regular quotient algebra. The concept of the passive collection being analog of involutive system of the nonlinear partial differential equations and Gröbner basis is introduced. Using special partitions of the algebra F and action of a semigroup on F, the sufficient conditions of passivity are obtained. The proof of the main theorem essentially differs from similar one for the differential equations and we obtain more general statement.

math.RA↗
arXiv2012

Application of the B-Determining Equations Method to One Problem of Free Turbulence

A three-dimensional model of the far turbulent wake behind a self-propelled body in a passively stratified medium is considered. The model is reduced to a system of ordinary differential equations by a similarity reduction and the B-determining equations method. The system of ordinary differential equations satisfying natural boundary conditions is solved numerically. The solutions obtained here are in close agreement with experimental data.

math-ph↗