695 citations · 1.8k across the 51 of their papers we have counts for
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Lax pair and first integrals for two of nonlinear coupled oscillators
Nikolay A. Kudryashov
The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing…
On the integrability conditions for a family of the Liénard-type equations
Nikolai A. Kudryashov, Dmitry I. Sinelshchikov
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…
New non-standard Lagrangians for the Liénard-type equations
Nikolai A. Kudryashov, Dmitry I. Sinelshchikov
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 admit…
On the criteria for integrability of the Liénard equation
Nikolai A. Kudryashov, Dmitry I. Sinelshchikov
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 ans…
Analytical solutions for problems of bubble dynamics
Nikolai A. Kudryashov, Dmitry I. Sinelshchikov
Recently, an asymptotic solution of the Rayleigh equation for an empty bubble in dimensions has been obtained. Here we give the closed--from general analytical solution of this…
Quasi-exact solutions of nonlinear differential equations
Nikolay A. Kudryashov, Mark B. Kochanov
The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters…
Analytical properties and exact solutions of the Lotka--Volterra competition system
Nikolay A. Kudryashov, Anastasia S. Zakharchenko
The Lotka--Volterra competition system with diffusion is considered. The Painlevé property of this system is investigated. Exact traveling wave solutions of the Lotka--Volterra com…
Logistic function as solution of many nonlinear differential equations
Nikolai A. Kudryashov, Mikhail A. Chmykhov
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 different…
Multi-particle dynamical systems and polynomials
Maria V. Demina, Nikolai A. Kudryashov
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding p…
Elliptic solutions in the Hénon - Heiles model
Maria V. Demina, Nikolai A. Kudryashov
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…
Elliptic solutions for a family of fifth order nonlinear evolution equations
Nikolay A. Kudryashov, Dmitry I. Sinelshchikov
The family of fifth order nonlinear evolution equations is studied. Some traveling wave elliptic solutions are found. The classification of these exact solutions is given.
Point vortices and classical orthogonal polynomials
Maria V. Demina, Nikolay A. Kudryashov
Stationary equilibria of point vortices with arbitrary choice of circulations in a background flow are studied. Differential equations satisfied by generating polynomials of vortex…
Meromorphic solutions of nonlinear ordinary differential equations
Nikolay A. Kudryashov
Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear…
Exact solutions of equations for the Burgers hierarchy
Nikolay A. Kudryashov, Dmitry I. Sinelshchikov
Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized C…
Self-similar solutions of the Burgers hierarchy
Nikolai A. Kudryashov
Self-similar solutions of the equations for the Burgers hierarchy are presented.
On elliptic solutions of nonlinear ordinary differential equations
Maria V. Demina, Nikolay A. Kudryashov
The generalized Bretherton equation is studied. The classification of the meromorphic traveling wave solutions for this equation is presented. All possible exact solutions of the g…
Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations
Maria V. Demina, Nikolay A. Kudryashov
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…
Vortices ans Polynomials: Nonuniqueness of the Adler-Moser polynomials for the Tkachenko equation
Maria V. Demina, Nikolay A. Kudryashov
Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of a system containing point vortices with arbitra…
Seven common errors in finding exact solutions of nonlinear differential equations
Nikolay A. Kudryashov
We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and cla…
Comment on: "New exact solutions for the Kawahara equation using Exp-function method"
Nikolai A. Kudryashov
Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com. Appl. Math. 233 (2009) 97--102] are analyzed. It is shown that all solutions do not satisfy the Kawahara eq…
A note on "New abundant solutions for the Burgers equation"
Nikolai A. Kudryashov, Dmitry I. Sinelshchikov
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) 5…
Comment on: "New exact solutions of the (3+1)-dimensional Burgers System" [Phys. Lett. A 373 (2009) 181]
Nikolai A. Kudryashov, Mikhail B. Soukharev
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 ge…
Remarks on rational solutions for the Korteveg - de Vries hierarchy
Nikolai A. Kudryashov
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 Kor…
Special polynomials associated with the K2 hierarchy
Nikolai A. Kudryashov
New special polynomials associated with the rational solutions of analogue to the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found.…