Summary_ (Исследование некоторых типов дифференциальных уравнений с сильной нелинейностью)

FEDERAL STATE BUDGET EDUCATIONAL INSTITUTION OF HIGHEREDUCATION«MIREA – RUSSIAN TECHNOLOGICAL UNIVERSITY»Rabinowitch Alexander SolomonovichInvestigation of some types of differential equations with strong nonlinearitySUMMARY of DISSERTATIONfor academic degree of Doctor of Sciencesin Applied Mathematics HSEMoscow – 20182GENERAL CHARACTERISTICS OF THE WORKThe relevance of the topic is related to the big role which the elaboration of newmathematical approaches plays to investigate poorly studied problems of mechanics andphysics that are described by nonlinear systems of differential equations.Although there exist a lot of mathematical methods in the theory of differential equations,nevertheless many problems in this field still remain to be solved which require a number ofnew methods.

In the first turn this concerns problems of mechanics and physics that arereduced to the analysis of systems of differential equations with strong nonlinearity. Thedissertation is devoted to investigations of a number of such problems. In it the followingquestions are studied:1) Investigation of the Navier-Stokes nonlinear differential equations describing flowsof a viscous incompressible fluid.The Navier-Stokes equations are basic equations of hydrodynamics and a large number ofworks are devoted to them. However, because of their substantial nonlinearity, numericalmethods prevail to study them and the number of the obtained analytical solutions is verysmall. That is why in the dissertation considerable attention is focused on the search ofparticular analytical solutions that could give a qualitative view of peculiarities of flows of aviscous incompressible fluid.

Besides, of special interest can be the case of high Reynoldsnumbers when the flow of a fluid could become turbulent and the case in which thephenomenon of cavitation takes place in the fluid.In the dissertation the following classes of axially symmetric solutions to the NavierStokes equations are studied:a) Investigation of classes of solutions to the Navier-Stokes equations which can berepresented in the form of power series in the radial coordinate with coefficients dependingon the time and axial coordinate.

Studying cases in which one can determine an explicit formof the coefficients and find conditions of convergence of these series and also cases in whichthey give solutions of closed forms.б) Investigation of special forms for the velocities and pressure in a viscousincompressible fluid which give particular analytical solutions to the three-dimensionalNavier-Stokes equations. Analysis of peculiarities of the solutions for high Reynoldsnumbers when a turbulent flow of a viscous fluid can occur.

Studying cases of viscous fluidflows when the phenomenon of cavitation can take place.2) Investigation of the nonlinear Yang-Mills equations to describe intensive non-AbelianfieldsAs is well-known, the Yang-Mills equations occupy a central place in description ofelectroweak and strong interactions and a large number of works are devoted to them.

Thefollowing works played important roles in studying them: non-Abelian spherically symmetricsolutions by T. Wu and Ch. Yang, G. ‘t Hooft and A Polyakov, the solution in the form of3non-Abelian plane waves by S. Coleman, which made an impact on the following works inthis field and also on this dissertation, nonstationary non-Abelian solutions by S. Matinyanand G Savvidy, investigations by R. Glassey and W. Strauss of the Cauchi problem for theYang-Mills equations and a large number of other worksIn the dissertation new classes of solutions to the Yang-Mills equations are sought. In itthe following types of solutions are obtained:а) Stationary and nonstationary spherically symmetric solutions to the Yang-Millsequations for classical field sources which present nonlinear generalizations of the classicalMaxwell solutions.b) Axially symmetric solutions to the Yang-Mills equations in the form of non-Abelianwaves propagating with the speed of light along some direction.c) Solutions to the Yang-Mills equations in the form of non-Abelian expanding wavespropagating with the speed of light.These solutions can be important to study nonlinear processes taking place in cosmicsources of Yang-Mills fields.3) Investigation of nonlinear dynamic equations for relativistic particles moving underthe action of Yukawa’s and Coulomb’s potentialsThe movement of a relativistic particle under the action of two central forces of theYukawa and Coulomb types is considered in the classical approximation.

In order to describeit, the classical Lagrangian for a particle moving in an electric field is used in which theYukawa potential is also introduced.This Lagrangian gives a system of nonlinear dynamic equations of the second order. Thesystem of equations is applied to a relativistic particle moving in some plane under the actionof central forces.

An investigation of the considered equations using the polar coordinates iscarried out and their first integral is determined. It is shown that under certain conditions theparticle can move in a closed orbit.The aim of the work is to study nonlinear differential equations which take place in anumber of physical problems.For this purpose, in the dissertation the following is made:- an investigation of axially symmetric solutions to the Navier-Stokes equations iscarried out for a viscous incompressible fluid that can be represented in the form of powerseries in the radial coordinate with coefficients depending on the time and axial coordinate;- a particular solution of a special form to the three-dimensional Navier-Stokes equationsis found and its peculiarities for high Reynolds numbers when turbulent flows of a viscousfluid can occur are considered;- a particular solution to the three-dimensional Navier-Stokes equations is found whichdescribes the case of fluid flows when cavitation takes place;4- stationary and nonstationary spherically symmetric solutions to the Yang-Millsequations with SU(2) symmetry for the classical sources of a special type are obtained andstudied;- non-Abelian wave solutions are found to describe radiations of cosmic sources of YangMills fields;- a nonlinear system of differential equations is obtained and studied which describes themovement of a relativistic particle in the central fields of Yukawa’s and Coulomb’spotentials.Scientific novelty consists in a mathematical analysis of nonlinear differential equationsthat take place in a number of physical problems.Some axially symmetric solutions to the Navier-Stokes equations for a viscousincompressible fluid are found in the form of power series in the radial coordinate withcoefficients depending on the time and axial coordinate.Particular solutions to the three-dimensional Navier-Stokes equations are found whichallow one to study some peculiarities of flows of a viscous fluid for high Reynolds numberswhen turbulence can occur and also when cavitation takes place.Some new types of stationary and nonstationary spherically symmetric solutions to theYang-Mills equations are obtained.Some axially symmetric non-Abelian solutions to the Yang-Mills equations and theirsolutions in the form of non-Abelian expanding waves are found.A system of nonlinear differential equations is considered and studied which describesthe movement of a relativistic particle under the action of the central fields of Yukawa’s andCoulomb’s types.Theoretical and practical importance of the work:Some new solutions to the Navier-Stokes equations obtained in the dissertation can beused for an analysis of flows of a viscous incompressible fluid for high Reynolds numberswhen turbulence can occur and also when cavitation takes place.New solutions to the Yang-Mills equations found in the work for spherically symmetricsolutions of a special type and for propagating non-Abelian waves can be applied to studynonlinear processes that take place in stars.The investigation carried out in it for a system of nonlinear differential equationsdescribing relativistic particles moving under the action of the central fields of the Yukawaand Coulomb types is important for a study of the dynamics of nucleons and antinucleons inthe fields of atomic nuclei.Scientific results presented for the defense of the thesis:1) New classes of axially symmetric solutions to the Navier-Stokes equations in the formof power series in the radial coordinate with coefficients depending on the time and axialcoordinate.52) New particular solutions of a special form to the three-dimensional Navier-Stokesequations and their peculiarity for high Reynolds numbers when a turbulent flow of a viscousfluid can occur and also when cavitation phenomena can take place.3) New classes of stationary and nonstationary spherically symmetric solutions to theYang-Mills equations.4) New types of axially symmetric non-Abelian wave solutions to the Yang-Millsequations and their solutions in the form of non-Abelian expanding waves.5) Investigation of a system of nonlinear differential equations for a description of themovement of a relativistic particle under the action of the central forces of the Yukawa andCoulomb types.Reliability of the obtained results.

The investigations carried out in the dissertationrely on a strict mathematical analysis of the studied nonlinear differential equations and thesolutions found for them.Approbation of the work. The dissertation materials were reported at the seminars ofthe Lomonosov Moscow State University (2000, 2009, 2012, 2017), People’s FriendshipUniversity of Russia (1998, 2003, 2005, 2012, 2017), Institute for Nuclear Research, RAS(2004), Steklov Mathematical Institute, RAS (2009, 2010), Prokhorov General PhysicsInstitute, RAS (2009), Russian Space Research Institute, RAS (2009), Keldysh Institute ofApplied Mathematics, RAS (2010, 2012), at 10-th, 11-th, 12-th, 16-th and 17-th Internationalschool-seminars on mathematical and theoretical physics (Kazan, 1998, 1999, 2000, 2004,2005), 11-th International conference of European Physical Society (London, 1999), 13-thLomonosov conference on physics of elementary particles (Moscow, 2007), 1-st Internationalconference on theoretical physics at Moscow State Open University (2011), 48-th, 49-th, 50th, 51-ой and 52-th All-Russia conferences on problems of particle physics, physics ofplasma and condensed media, optoelectronics at People’s Friendship University of Russia(Moscow, 2012, 2013, 2014, 2015, 2016, 2017) and at Ya.