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Nikolai A. Kudryashov

Publications and source records attributed to Nikolai A. Kudryashov.

At least 19 recordsLinked to original sources
arXiv2016

On the integrability conditions for a family of the Liénard-type equations

We study a family of Liénard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this equations family. To this end we use an approach, which is bases on application of nonlocal transformations. By studying connections between this family of Liénard--type equations and type III Painlevé--Gambier equations, we obtain four new integrability criteria. We illustrate our results by providing examples of some integrable Liénard--type equations. We also discuss relationships between linearizability via nonlocal transformations of this family of Liénard--type equations and other integrability conditions for this family of equations.

nlin.SI↗
arXiv2016

New non-standard Lagrangians for the Liénard-type equations

Liénard-type equations are used for the description of various phenomena in physics and other fields of science. Here we find a new family of the Liénard-type equations which admits a non-standard autonomous Lagrangian. As a by-product we obtain autonomous first integrals for each member of this family of equations. We also show that some of the previously known conditions for the existence of a non-standard Lagrangian for the Liénard-type equations follow from the linearizability of the corresponding equation via nonlocal transformations.

nlin.SI↗
arXiv2016

On the criteria for integrability of the Liénard equation

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of previously known criteria for integrability of the Liénard equation as a straightforward consequences of our approach's application. We illustrate our results by several new examples of integrable Liénard equations.

nlin.SI↗
arXiv2016

Periodic structures described by the perturbed Burgers-Korteweg-de Vries equation

We study the perturbed Burgers-Korteweg-de Vries equation. This equation can be used for the description of nonlinear waves in a liquid with gas bubbles and for the description of nonlinear waves on a fluid layer flowing down an inclined plane. We investigate the integrability of this equation using the Painlevé approach. We show that the perturbed Burgers-Korteweg-de Vries equation does not belong to the class of integrable equations. Classical and nonclassical symmetries admitted by this equation and corresponding symmetry reductions are constructed. New types of periodic analytical structures described by the Burgers-Korteweg-de Vries equation are found.

nlin.PS↗
arXiv2016

Analytical solutions for problems of bubble dynamics

Recently, an asymptotic solution of the Rayleigh equation for an empty bubble in $N$ dimensions has been obtained. Here we give the closed--from general analytical solution of this equation. We also find the general solution of the Rayleigh equation in $N$ dimensions for the case of a gas--filled hyperspherical bubble. In addition, we include a surface tension into consideration.

nlin.SI↗
arXiv2014

Logistic function as solution of many nonlinear differential equations

The logistic function is shown to be solution of the Riccati equation, some second-order nonlinear ordinary differential equations and many third-order nonlinear ordinary differential equations. The list of the differential equations having solution in the form of the logistic function is presented. The simple method of finding exact solutions of nonlinear partial differential equations (PDEs) is introduced. The essence of the method is based on comparison of nonlinear differential equations obtained from PDEs with standard differential equations having solution in the form of the logistic function. The wide application of the logistic function for finding exact solutions of nonlinear differential equations is demonstrated.

nlin.SI↗
arXiv2014

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is described. The method enables one to integrate a wide class of polynomial multi--particle dynamical systems. The general solutions of certain dynamical systems related to linear second--order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived. Our approach is also applicable to dynamical systems that are not multi--particle by their nature but that can be regarded as multi--particle (for example, the Darboux--Halphen system and its generalizations). A wide class of two and three--particle polynomial dynamical systems is integrated.

nlin.SI↗
arXiv2013

Extended equation for description of nonlinear waves in liquid with gas bubbles

Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for long weakly nonlinear waves taking into consideration liquid viscosity, inter--phase heat transfer and surface tension. Additional conditions for the parameters of the equation are determined for integrability of the mathematical model. The transformation for linearization of the nonlinear equation is presented too. Some exact solutions of the nonlinear equation are found for integrable and non--integrable cases. The nonlinear waves described by the nonlinear equation are numerically investigated.

nlin.PS↗
arXiv2012

Elliptic solutions in the Hénon - Heiles model

Equations of motion corresponding to the Hénon - Heiles system are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential equations is described. New families of elliptic solutions of a fourth--order equation related to the Hénon - Heiles system are obtained. A classification of elliptic solutions up to the sixth order inclusively is presented.

nlin.SI↗
arXiv2012

A Lagrangian Description of the Higher-Order Painlevé Equations

We derive the Lagrangians of the higher-order Painlevé equations using Jacobi's last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painlevé test and satisfy the conditions stated by Jurá$\check{s}$, (Acta Appl. Math. 66 (2001) 25--39), thus allowing for a Lagrangian description.

nlin.SI↗
arXiv2012

Thermal regimes of high burn-up nuclear fuel rod

The temperature distribution in the nuclear fuel rods for high burn-up is studied. We use the numerical and analytical approaches. It is shown that the time taken to have the stationary thermal regime of nuclear fuel rod is less than one minute. We can make the inference that the behavior of the nuclear fuel rod can be considered as a stationary task. Exact solutions of the temperature distribution in the fuel rods in the stationary case are found. Thermal regimes of high burn-up the nuclear fuel rods are analyzed.

physics.chem-ph↗
arXiv2011

On one of methods for finding exact solutions of nonlinear differential equations

One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and nonlinear ordinary differential equation of the seven order. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations. Merits and demerits of the method are discussed.

nlin.SI↗
arXiv2009

Comment on: "New exact solutions of the (3+1)-dimensional Burgers System" [Phys. Lett. A 373 (2009) 181]

We demonstrate that all exact solutions of the Riccati equation by Dai andWang [C.-Q. Dai, Y.-Y.Wang, Phys. Lett. A 373 (2009) 181-187] are not new and cannot be new because the general solution of this equation was obtained more than one century ago. Moreover we show that some "new solutions" by Dai and Wang of the Riccati equation do not satisfy this equation. We also illustrate that the authors did not obtain any new solutions for solution of the (3+1)-dimensional Burgers system.

nlin.SI↗
arXiv2009

A note on "New abundant solutions for the Burgers equation"

Salas, Gomez and Heranandez [A.Y. Salas S., C.A. Gomez S., J.E.C Hernandez, New abundant solutions for tha Burgers equation, Computers and Mathematics with Applications 58 (2009) 514 -520] presented 70 "new exact solutions" of a "generalized version" of the Burgers equation. In this comment we show that all 70 solutions by these authors are not new and cannot be new.

nlin.SI↗
arXiv2007

Remarks on rational solutions for the Korteveg - de Vries hierarchy

Differential equations for the special polynomials associated with the rational solutions of the second Painleve hierarchy are introduced. It is shown rational solutions of the Korteveg - de Vries hierarchy can be found taking the Yablonskii - Vorob'ev polynomials into account. Special polynomials associated with rational solutions of the Korteveg - de Vries hierarchy are presented.

nlin.SI↗