Nikolai Kudryashov
Publications:
Kudryashov N. A.
Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies
2021, vol. 26, no. 3, pp. 271292
Abstract
Selfsimilar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are
considered. Algorithms for constructing a Lax pair
for equations of these hierarchies are presented. Lax pairs for ordinary differential
equations of the fifth, seventh and eleventh orders
corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given.
The Lax pairs allow us to solve these equations by means of the inverse
monodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is
demonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via
the transcendents of the $K_1$ and $K_2$ hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera
hierarchies are given. Special polynomials associated with the selfsimilar reductions of
the Kupershmidt and Sawada – Kotera hierarchies are presented.
Rational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials.

Kudryashov N. A.
Highly Dispersive Optical Solitons of an Equation with Arbitrary Refractive Index
2020, vol. 25, no. 6, pp. 537543
Abstract
A nonlinear fourthorder differential equation with arbitrary refractive index for
description of the pulse propagation in an optical fiber is considered. The Cauchy problem for
this equation cannot be solved by the inverse scattering transform and we look for solutions
of the equation using the traveling wave reduction. We present a novel method for finding
soliton solutions of nonlinear evolution equations. The essence of this method is based on the
hypothesis about the possible type of an auxiliary equation with an already known solution. This
new auxiliary equation is used as a basic equation to look for soliton solutions of the original
equation. We have found three forms of soliton solutions of the equation at some constraints
on parameters of the equation.

Kudryashov N. A.
Rational Solutions of Equations Associated with the Second Painlevé Equation
2020, vol. 25, no. 3, pp. 273280
Abstract
Nonlinear differential equations associated with the second Painlevé equation are considered. Transformations for solutions of the singular manifold equation are presented. It is shown that rational solutions of the singular manifold equation are determined by means of the Yablonskii – Vorob’ev polynomials. It is demonstrated that rational solutions for some differential equations are also expressed via the Yablonskii – Vorob’ev polynomials.

Kudryashov N. A.
Lax Pairs and Special Polynomials Associated with Selfsimilar Reductions of Sawada – Kotera and Kupershmidt Equations
2020, vol. 25, no. 1, pp. 5977
Abstract
Selfsimilar reductions of the Sawada – Kotera and Kupershmidt equations are
studied. Results of Painlevé's test for these equations are given. Lax pairs for solving the
Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions
of the Sawada – Kotera and Kupershmidt equations expressed via the first Painlevé equation
are presented. Exact solutions of the Sawada – Kotera and Kupershmidt equations by means
of general solution for the first member of $K_2$ hierarchy are given. Special polynomials for
expressions of rational solutions for the equations considered are introduced. The differentialdifference
equations for finding special polynomials corresponding to the Sawada – Kotera and
Kupershmidt equations are found. Nonlinear differential equations of sixth order for special
polynomials associated with the Sawada – Kotera and Kupershmidt equations are obtained.
Lax pairs for nonlinear differential equations with special polynomials are presented. Rational
solutions of the selfsimilar reductions for the Sawada – Kotera and Kupershmidt equations are
given.

Kudryashov N. A., Safonova D. V., Biswas A.
Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation
2019, vol. 24, no. 6, pp. 607614
Abstract
This paper considers the Radhakrishnan – Kundu – Laksmanan (RKL) equation to
analyze dispersive nonlinear waves in polarizationpreserving fibers. The Cauchy problem for
this equation cannot be solved by the inverse scattering transform (IST) and we look for exact
solutions of this equation using the traveling wave reduction. The Painlevé analysis for the
traveling wave reduction of the RKL equation is discussed. A first integral of traveling wave
reduction for the RKL equation is recovered. Using this first integral, we secure a general
solution along with additional conditions on the parameters of the mathematical model. The
final solution is expressed in terms of the Weierstrass elliptic function. Periodic and solitary
wave solutions of the RKL equation in the form of the traveling wave reduction are presented
and illustrated.

Kudryashov N. A.
Rational and Special Solutions for Some Painlevé Hierarchies
2019, vol. 24, no. 1, pp. 90100
Abstract
A selfsimilar 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 selfsimilar 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 selfsimilar 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.

Safonova D. V., Demina M. V., Kudryashov N. A.
Stationary Configurations of Point Vortices on a Cylinder
2018, vol. 23, no. 5, pp. 569579
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.

Kudryashov N. A.
Exact Solutions and Integrability of the Duffing–Van der Pol Equation
2018, vol. 23, no. 4, pp. 471479
Abstract
The forcefree 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 wellknown 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 secondorder ordinary differential equations.

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. 257272
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.

Kudryashov N. A.
Asymptotic and Exact Solutions of the FitzHugh–Nagumo Model
2018, vol. 23, no. 2, pp. 152160
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 secondorder 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 secondorder 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.

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. 266271
Abstract
The fourthorder 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.

Kudryashov N. A., Sinelshchikov D. I.
On the Integrability Conditions for a Family of Liénardtype Equations
2016, vol. 21, no. 5, pp. 548555
Abstract
We study a family of Liénardtype 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 Liénardtype equations and type III Painlevé–Gambier equations, we obtain four new integrability criteria. We illustrate our results by providing examples of some integrable Liénardtype equations. We also discuss relationships between linearizability via nonlocal transformations of this family of Liénardtype equations and other integrability conditions for this family of equations.

Demina M. V., Kudryashov N. A.
Multiparticle Dynamical Systems and Polynomials
2016, vol. 21, no. 3, pp. 351366
Abstract
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multiparticle dynamical system
by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multiparticle dynamical systems. The general solutions of certain dynamical systems related to linear secondorder partial differential equations are found. As a byproduct of our results, new families of orthogonal polynomials are derived.

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. 486496
Abstract
The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear secondorder 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.

Kudryashov N. A.
Analytical Solutions of the Lorenz System
2015, vol. 20, no. 2, pp. 123133
Abstract
The Lorenz system is considered. The Painlevé test for the thirdorder 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.

Kudryashov N. A., Sinelshchikov D. I.
Special Solutions of a Highorder Equation for Waves in a Liquid with Gas Bubbles
2014, vol. 19, no. 5, pp. 576585
Abstract
A fifthorder 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 highorder analogs is discussed.

Borisov A. V., Kudryashov N. A.
Paul Painlevé and His Contribution to Science
2014, vol. 19, no. 1, pp. 119
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 secondorder 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.

Kudryashov N. A.
Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies
2014, vol. 19, no. 1, pp. 4863
Abstract
It is well known that the selfsimilar 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 selfsimilar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higherorder Painlevé hierarchies introduced more than ten years ago.

Demina M. V., Kudryashov N. A.
Relative Equilibrium Configurations of Point Vortices on a Sphere
2013, vol. 18, no. 4, pp. 344355
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.

Demina M. V., Kudryashov N. A.
Point Vortices and Classical Orthogonal Polynomials
2012, vol. 17, no. 5, pp. 371384
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.

Demina M. V., Kudryashov N. A.
Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations
2011, vol. 16, no. 6, pp. 562576
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.

Kudryashov N. A., Soukharev M. B.
Popular Ansatz Methods and Solitary Wave Solutions of the Kuramoto–Sivashinsky Equation
2009, vol. 14, no. 3, pp. 407419
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.

Kudryashov N. A.
Solitary and Periodic Solutions of the Generalized Kuramoto–Sivashinsky Equation
2008, vol. 13, no. 3, pp. 234238
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.

Efimova O. Y., Kudryashov N. A.
Power Expansions for the SelfSimilar Solutions of the Modified Sawada–Kotera Equation
2007, vol. 12, no. 2, pp. 198218
Abstract
The fourthorder ordinary differential equation that denes the selfsimilar solutions of the Kaup–Kupershmidt and Sawada–Kotera equations is studied. This equation belongs to the class of fourthorder analogues of the Painlevé equations. All the power and nonpower 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.
