2,213 research outputs found
Explicit expressions for meromorphic solution of autonomous nonlinear ordinary differential equations
Meromorphic solutions of autonomous nonlinear ordinary differential equations
are studied. An algorithm for constructing meromorphic solutions in explicit
form is presented. General expressions for meromorphic solutions (including
rational, periodic, elliptic) are found for a wide class of autonomous
nonlinear ordinary differential equations
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.Comment: 14 pages. This paper was sent to the Communications in Nonlinear
Science and Numerical Simulatio
Exact solutions of the generalized equations
Family of equations, which is the generalization of the equation, is
considered. Periodic wave solutions for the family of nonlinear equations are
constructed
Power expansions for solution of the fourth-order analog to the first Painlev\'{e} equation
One of the fourth-order analog to the first Painlev\'{e} equation is studied.
All power expansions for solutions of this equation near points and
are found by means of the power geometry method. The exponential
additions to the expansion of solution near are computed. The
obtained results confirm the hypothesis that the fourth-order analog of the
first Painlev\'{e} equation determines new transcendental functions.Comment: 28 pages, 5 figure
Painleve property and the first integrals of nonlinear differential equations
Link between the Painleve property and the first integrals of nonlinear
ordinary differential equations in polynomial form is discussed. The form of
the first integrals of the nonlinear differential equations is shown to
determine by the values of the Fuchs indices. Taking this idea into
consideration we present the algorithm to look for the first integrals of the
nonlinear differential equations in the polynomial form. The first integrals of
five nonlinear ordinary differential equations are found. The general solution
of one of the fourth ordinary differential equations is given.Comment: 22 page
Meromorphic exact solutions of the generalized Bretherton equation
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 generalized Brethenton equation are given
Exact solutions of equations for the Burgers hierarchy
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 Cole - Hopf transformation
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
Relations for zeros of special polynomials associated to the Painleve equations
A method for finding relations for the roots of polynomials is presented. Our
approach allows us to get a number of relations for the zeros of the classical
polynomials and for the roots of special polynomials associated with rational
solutions of the Painleve equations. We apply the method to obtain the
relations for the zeros of several polynomials. They are: the Laguerre
polynomials, the Yablonskii - Vorob'ev polynomials, the Umemura polynomials,
the Ohyama polynomials, the generalized Okamoto polynomials, and the
generalized Hermite polynomials. All the relations found can be considered as
analogues of generalized Stieltjes relations.Comment: 17 pages, 5 figure
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.Comment: 42 page
- …