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Nikolay A. Kudryashov

Publications and source records attributed to Nikolay A. Kudryashov.

At least 37 records · Page 2Linked to original sources
arXiv2012

Equation for three-dimensional nonlinear waves in liquid with gas bubbles

Nonlinear waves in a liquid containing gas bubbles are considered in the three-dimensional case. Nonlinear evolution equation is given for description of long nonlinear pressure waves. It is shown that in the general case the equation is not integrable. Some exact solutions for the nonlinear evolution equation are presented. Application of the Hirota method is illustrated for finding multi-soliton solutions for the nonintegrable evolution equation in the three-dimensional case. The stability of the one-dimensional solitary waves is investigated. It is shown that the one-dimensional solitary waves are stable to transverse perturbations.

nlin.PS↗
arXiv2012

Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices with arbitrary choice of circulations in a background flow are studied. 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.

nlin.SI↗
arXiv2011

Meromorphic solutions of nonlinear ordinary differential equations

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 ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations.

nlin.SI↗
arXiv2011

Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.

nlin.PS↗
arXiv2011

Exact solutions of the Swift-Hohenberg equation with dispersion

The Swift-Hohenberg equation with dispersion is considered. Traveling wave solutions of the Swift-Hohenberg equation with dispersion are presented. The classification of these solutions is given. It is shown that the Swift-Hohenberg equation without dispersion has only stationary meromorphic solution.

nlin.PS↗
arXiv2011

Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer

Nonlinear waves are studied in a mixture of liquid and gas bubbles. Influence of viscosity and heat transfer is taken into consideration on propagation of the pressure waves. Nonlinear evolution equations of the second and the third order for describing nonlinear waves in gas-liquid mixtures are derived. Exact solutions of these nonlinear evolution equations are found. Properties of nonlinear waves in a liquid with gas bubbles are discussed.

nlin.PS↗
arXiv2011

From Laurent Series to Exact Meromorphic Solutions: the Kawahara equation

Nonlinear waves are studied in a mixture of liquid and gas bubbles. Influence of viscosity and heat transfer is taken into consideration on propagation of the pressure waves. Nonlinear evolution equations of the second and the third order for describing nonlinear waves in gas-liquid mixtures are derived. Exact solutions of these nonlinear evolution equations are found. Properties of nonlinear waves in a liquid with gas bubbles are discussed.

nlin.SI↗
arXiv2011

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

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 -2Gamma 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.

nlin.SI↗
arXiv2011

Vortices ans Polynomials: Nonuniqueness of the Adler-Moser polynomials for the Tkachenko equation

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 arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is obtained that the Adler - Moser polynomial are not unique polynomial solutions of the Tkachenko equation. A generalization of the Tkachenko equation to the case of translating relative equilibria is derived. It is shown that the generalization of the Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers.

nlin.SI↗
arXiv2010

Be careful with the Exp-function method

An application of the Exp-function method to search for exact solutions of nonlinear differential equations is analyzed. Typical mistakes of application of the Exp-function method are demonstrated. We show it is often required to simplify the exact solutions obtained. Possibilities of the Exp-function method and other approaches in mathematical physics are discussed. The application of the singular manifold method for finding exact solutions of the Fitzhugh - Nagumo equation is illustrated. The modified simple equation method is introduced. This approach is used to look for exact solutions of the generalized Korteweg - de Vries equation.

nlin.SI↗
arXiv2010

Seven common errors in finding exact solutions of nonlinear differential equations

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 classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.

nlin.SI↗
arXiv2010

On Completely Integrability Systems of Differential Equations

In this note we discuss the approach which was given by Wazwaz for the proof of the complete integrability to the system of nonlinear differential equations. We show that his method presented in [Wazwaz A.M. Completely integrable coupled KdV and coupled KP systems, Commun Nonlinear Sci Simulat 15 (2010) 2828-2835] is incorrect.

nlin.SI↗
arXiv2010

A Note on "Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation"

We discuss the recent paper by Inan and Ugurlu [Inan I.E., Ugurlu Y., Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation, Appl. Math. Comp. 217 (2010) 1294 -- 1299]. We demonstrate that all exact solutions of fifth order KdV equation and modified Burgers equation by Inan and Ugurlu are trivial solutions that are reduced to constants. Moreover, we show exact solutions of the fifth -- order equation studied by Inan and Ugurlu cannot be found by the Exp-function method.

nlin.SI↗