Related papers: Singularity methods for meromorphic solutions of d…
We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…
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,…
New method is presented to look for exact solutions of nonlinear differential equations. Two basic ideas are at the heart of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another…
Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a…
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…
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.
In this paper we shall study differential equations in the complex domain. The method of indeterminate coefficients and the majorant method lead to a proof of the existence and uniqueness of meromorphic solution of differential equations.…
Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first…
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
A meromorphic solution of a complex linear differential equation (with meromorphic coefficients) for which the value zero is the only possible finite deficient/deviated value is called a standard solution. Conditions for the existence and…
New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for…
In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by…
We determine all cases when there exists a meromorphic solution of the third order ODE describing traveling waves solutions of the Kuramoto-Sivashinsky equation. It turns out that there are no other meromorphic solutions besides those…
The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can…
A special class of autonomous algebraic differential equations is studied. No equations in the class have any entire transcendental solutions. In a sense, for almost all equations in the class, transcendental meromorphic solutions can also…
Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a…
We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
This paper is devoted to overview of the authors works for numerical solution of singular integral equations (SIE), polysingular integral equations and multi-dimensional singular integral equations of the second kind. The authors…