0
2013
Impact Factor

Nikolai Kudryashov Kashirskoe sh. 31, Moscow 115409, Russia
Moscow Engineering and Physics Institute

Publications:

 Kudryashov N. A. Rational and Special Solutions for Some Painlevé Hierarchies 2019, vol. 24, no. 1, pp.  90-100 Abstract A self-similar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented. This Lax pair can be used to solve the Cauchy problem for the studied hierarchy. It is shown that special solutions of the self-similar reduction of the KdV hierarchy are determined via the transcendents of the first Painlevé hierarchy. Rational solutions are expressed by means of the Yablonskii–Vorob’ev polynomials. The map of the solutions for the second Painlevé hierarchy into the solutions for the self-similar reduction of the KdV hierarchy is illustrated using the Miura transformation. Lax pairs for equations of the hierarchy for the Yablonskii–Vorob’ev polynomial are discussed. Special solutions to the hierarchy for the Yablonskii–Vorob’ev polynomials are given. Some other hierarchies with properties of the Painlevé hierarchies are presented. The list of nonlinear differential equations whose general solutions are expressed in terms of nonclassical functions is extended. Keywords: self-similar reduction, KdV hierarchy, Painlevé hierarchy, Painlevé transcendent, transformation Citation: Kudryashov N. A.,  Rational and Special Solutions for Some Painlevé Hierarchies, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 90-100 DOI:10.1134/S1560354719010052
 Safonova D. V., Demina M. V., Kudryashov N. A. Stationary Configurations of Point Vortices on a Cylinder 2018, vol. 23, no. 5, pp.  569-579 Abstract In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation $\Gamma_1$ and $\Gamma_2$ $(\Gamma_2 = -\mu\Gamma_1)$ are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained. Keywords: point vortices, stagnation points, stationary configuration, vortices on a cylinder, polynomial solution of differential equation Citation: Safonova D. V., Demina M. V., Kudryashov N. A.,  Stationary Configurations of Point Vortices on a Cylinder, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 569-579 DOI:10.1134/S1560354718050064
 Kudryashov N. A. Exact Solutions and Integrability of the Duffing–Van der Pol Equation 2018, vol. 23, no. 4, pp.  471-479 Abstract The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations. Keywords: Duffing–Van der Pol oscillator, Painlevé test, exact solution, truncated expansion, singular manifold, general solution Citation: Kudryashov N. A.,  Exact Solutions and Integrability of the Duffing–Van der Pol Equation, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 471-479 DOI:10.1134/S156035471804007X
 Garashchuk I. R., Sinelshchikov D. I., Kudryashov N. A. Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall 2018, vol. 23, no. 3, pp.  257-272 Abstract Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications. Keywords: contrast agent, dynamical system, nonlinear dynamics, dynamical chaos, multistability, coexisting attractors Citation: Garashchuk I. R., Sinelshchikov D. I., Kudryashov N. A.,  Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall, Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 257-272 DOI:10.1134/S1560354718030036
 Kudryashov N. A. Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model 2018, vol. 23, no. 2, pp.  152-160 Abstract The standard FitzHugh–Nagumo model for description of impulse from one neuron to another is considered. The system of equations is transformed to a nonlinear second-order ordinary differential equation. It is shown that the differential equation does not pass the Painlevé test in the general case and the general solution of this equation does not exist. The simplest solutions of the system of equations are found. The second-order differential equation is transformed to another asymptotic equation with the general solution expressed via the Jacobi elliptic function. This transformation allows us to obtain the asymptotic solutions of the FitzHugh–Nagumo model. The perturbed FitzHugh–Nagumo model is studied as well. Taking into account the simplest equation method, the exact solutions of the perturbed system of equations are found. The asymptotic solutions of the perturbed model are presented too. The application of the exact solutions for construction of the neural networks is discussed. Keywords: neuron, FitzHugh–Nagumo model, system of equations, Painelevé test, exact solution Citation: Kudryashov N. A.,  Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model, Regular and Chaotic Dynamics, 2018, vol. 23, no. 2, pp. 152-160 DOI:10.1134/S1560354718020028
 Kudryashov N. A., Gaur I. Y. Weak Nonlinear Asymptotic Solutions for the Fourth Order Analogue of the Second Painlevé Equation 2017, vol. 22, no. 3, pp.  266-271 Abstract The fourth-order analogue of the second Painlevé equation is considered. The monodromy manifold for a Lax pair associated with the $P_2^2$ equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays $\phi = \frac{2}{5}\pi(2n+1)$ on the complex plane have been found by the isomonodromy deformations technique. Keywords: $P^2_2$ equation, isomonodromy deformations technique, special functions, Painlevé transcendents Citation: Kudryashov N. A., Gaur I. Y.,  Weak Nonlinear Asymptotic Solutions for the Fourth Order Analogue of the Second Painlevé Equation, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 266-271 DOI:10.1134/S1560354717030066
 Kudryashov N. A., Sinelshchikov D. I. On the Integrability Conditions for a Family of Liénard-type Equations 2016, vol. 21, no. 5, pp.  548-555 Abstract 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 travelingwave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this family of equations. To this end we use an approach which is based on the 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. Keywords: Liénard-type equation, nonlocal transformations, closed-form solution, general solution, Painlevé–Gambier equations Citation: Kudryashov N. A., Sinelshchikov D. I.,  On the Integrability Conditions for a Family of Liénard-type Equations, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 548-555 DOI:10.1134/S1560354716050063
 Demina M. V., Kudryashov N. A. Multi-particle Dynamical Systems and Polynomials 2016, vol. 21, no. 3, pp.  351-366 Abstract 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 partial differential equations is introduced. 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. Keywords: multi-particle dynamical systems, polynomial solutions of partial differential equations, orthogonal polynomials Citation: Demina M. V., Kudryashov N. A.,  Multi-particle Dynamical Systems and Polynomials, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 351-366 DOI:10.1134/S1560354716030072
 Kudryashov N. A., Sinelshchikov D. I. On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions 2015, vol. 20, no. 4, pp.  486-496 Abstract The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear second-order differential equation has been discussed. Here we show that the whole family of quadratic Lienard equations can be transformed into an equation for the elliptic functions. We demonstrate that this connection can be useful for finding explicit forms of general solutions of the quadratic Lienard equation. We provide several examples of application of our approach. Keywords: quadratic lienard equation, elliptic functions, nonlocal transformations, general solution Citation: Kudryashov N. A., Sinelshchikov D. I.,  On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 486-496 DOI:10.1134/S1560354715040073
 Kudryashov N. A. Analytical Solutions of the Lorenz System 2015, vol. 20, no. 2, pp.  123-133 Abstract The Lorenz system is considered. The Painlevé test for the third-order equation corresponding to the Lorenz model at $\sigma \ne 0$ is presented. The integrable cases of the Lorenz system and the first integrals for the Lorenz system are discussed. The main result of the work is the classification of the elliptic solutions expressed via the Weierstrass function. It is shown that most of the elliptic solutions are degenerated and expressed via the trigonometric functions. However, two solutions of the Lorenz system can be expressed via the elliptic functions. Keywords: Lorenz system, Painlevé property, Painlevé test, analytical solutions, elliptic solutions Citation: Kudryashov N. A.,  Analytical Solutions of the Lorenz System, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 123-133 DOI:10.1134/S1560354715020021
 Kudryashov N. A., Sinelshchikov D. I. Special Solutions of a High-order Equation for Waves in a Liquid with Gas Bubbles 2014, vol. 19, no. 5, pp.  576-585 Abstract A fifth-order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling wave solutions are constructed. Connection of selfsimilar solutions with Painlevé transcendents and their high-order analogs is discussed. Keywords: waves in a liquid with gas bubbles, evolution equations, exact solutions, meromorphic solutions, fifth-order KdV equation, Kaup–Kupershmidt equation Citation: Kudryashov N. A., Sinelshchikov D. I.,  Special Solutions of a High-order Equation for Waves in a Liquid with Gas Bubbles, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 576-585 DOI:10.1134/S1560354714050050
 Borisov A. V., Kudryashov N. A. Paul Painlevé and His Contribution to Science 2014, vol. 19, no. 1, pp.  1-19 Abstract The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the $N$-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed. Keywords: mathematician, politician, Painlevé equations; Painlevé transcendents; Painlevé paradox Citation: Borisov A. V., Kudryashov N. A.,  Paul Painlevé and His Contribution to Science, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 1-19 DOI:10.1134/S1560354714010018
 Kudryashov N. A. Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies 2014, vol. 19, no. 1, pp.  48-63 Abstract It is well known that the self-similar solutions of the Korteweg–de Vries equation and the modified Korteweg–de Vries equation are expressed via the solutions of the first and second Painlevé equations. In this paper we solve this problem for all equations from the Korteveg–de Vries, modified Korteweg–de Vries, Kaup–Kupershmidt, Caudrey–Dodd–Gibbon and Fordy–Gibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painlevé hierarchies introduced more than ten years ago. Keywords: Painlevé equation, Painlevé transcendent, Korteweg–de Vries hierarchy, modified Korteveg–de Vries hierarchy, Kaup–Kupershmidt hierarchy, Caudrey–Dodd–Cibbon hierarchy Citation: Kudryashov N. A.,  Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 48-63 DOI:10.1134/S1560354714010043
 Demina M. V., Kudryashov N. A. Relative Equilibrium Configurations of Point Vortices on a Sphere 2013, vol. 18, no. 4, pp.  344-355 Abstract The problem of constructing and classifying equilibrium and relative equilibrium configurations of point vortices on a sphere is studied. A method which enables one to find any such configuration is presented. Configurations formed by the vortices placed at the vertices of Platonic solids are considered without making the assumption that the vortices possess equal in absolute value circulations. Several new configurations are obtained. Keywords: point vortices, sphere, relative equilibrium, fixed equilibrium, Platonic solids Citation: Demina M. V., Kudryashov N. A.,  Relative Equilibrium Configurations of Point Vortices on a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 344-355 DOI:10.1134/S1560354713040023
 Demina M. V., Kudryashov N. A. Point Vortices and Classical Orthogonal Polynomials 2012, vol. 17, no. 5, pp.  371-384 Abstract Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials. Keywords: point vortices, special polynomials, classical orthogonal polynomials Citation: Demina M. V., Kudryashov N. A.,  Point Vortices and Classical Orthogonal Polynomials, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 371-384 DOI:10.1134/S1560354712050012
 Demina M. V., Kudryashov N. A. Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations 2011, vol. 16, no. 6, pp.  562-576 Abstract Rational solutions and special polynomials associated with the generalized $K_2$ hierarchy are studied. This hierarchy is related to the Sawada–Kotera and Kaup–Kupershmidt equations and some other integrable partial differential equations including the Fordy–Gibbons equation. Differential–difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations $\Gamma$ and $−2\Gamma$ is established. Properties of the polynomials are studied. Differential–difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found. Keywords: point vortices, special polynomials, generalized $K_2$ hierarchy, Sawada–Kotera equation, Kaup–Kupershmidt equation, Fordy–Gibbons equation Citation: Demina M. V., Kudryashov N. A.,  Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 562-576 DOI:10.1134/S1560354711060025
 Kudryashov N. A., Soukharev  M. B. Popular Ansatz Methods and Solitary Wave Solutions of the Kuramoto–Sivashinsky Equation 2009, vol. 14, no. 3, pp.  407-419 Abstract Some methods to look for exact solutions of nonlinear differential equations are discussed. It is shown that many popular methods are equivalent to each other. Several recent publications with "new" solitary wave solutions for the Kuramoto–Sivashinsky equation are analyzed. We demonstrate that all these solutions coincide with the known ones. Keywords: Kuramoto–Sivashinsky equation, $(G'/G)$-method, Tanh-method, Exp-function method Citation: Kudryashov N. A., Soukharev  M. B.,  Popular Ansatz Methods and Solitary Wave Solutions of the Kuramoto–Sivashinsky Equation, Regular and Chaotic Dynamics, 2009, vol. 14, no. 3, pp. 407-419 DOI:10.1134/S1560354709030046
 Kudryashov N. A. Solitary and Periodic Solutions of the Generalized Kuramoto–Sivashinsky Equation 2008, vol. 13, no. 3, pp.  234-238 Abstract The generalized Kuramoto–Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented. Keywords: exact solution, nonlinear differential equation, Kuramoto–Sivashinsky equation Citation: Kudryashov N. A.,  Solitary and Periodic Solutions of the Generalized Kuramoto–Sivashinsky Equation, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 234-238 DOI:10.1134/S1560354708030088
 Efimova O. Y., Kudryashov N. A. Power Expansions for the Self-Similar Solutions of the Modified Sawada–Kotera Equation 2007, vol. 12, no. 2, pp.  198-218 Abstract The fourth-order ordinary differential equation that denes the self-similar solutions of the Kaup–Kupershmidt and Sawada–Kotera equations is studied. This equation belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points $z = 0$, $z = \infty$ and near an arbitrary point $z = z_0$ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined. Keywords: Kaup–Kupershmidt equation, Sawada–Kotera equation, fourth-order analogue of the second Painlevé equation, power geometry methods, asymptotic forms, power expansions Citation: Efimova O. Y., Kudryashov N. A.,  Power Expansions for the Self-Similar Solutions of the Modified Sawada–Kotera Equation, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 198-218 DOI:10.1134/S1560354707020062