Related papers: Construction of Special Solutions for Nonintegrabl…
The Painleve test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. To find the…
The generalized Henon-Heiles system has been considered. In two nonintegrable cases with the help of the Painleve test new special solutions have been found as Laurent series, depending on three parameters. The obtained series converge in…
The Henon-Heiles system in the general form has been considered. In a nonintegrable case with the help of the Painleve test new solutions have been found as formal Laurent or Puiseux series, depending on three parameters. One of parameters…
The H\'enon--Heiles system in the general form is studied. In a nonintegrable case new solutions have been found as formal Laurent series, depending on three parameters. One of parameters determines a location of the singularity point,…
The Painleve and weak Painleve conjectures have been used widely to identify new integrable nonlinear dynamical systems. The calculation of the integrals relies though on methods quite independent from Painlev\'e analysis. This paper…
The automation of the traditional Painleve test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of ordinary and partial differential equations which may be parameterized by arbitrary…
The generalized Henon-Heiles system with an additional nonpolynomial term is considered. In two nonintegrable cases new two-parameter solutions have been obtained in terms of elliptic functions. These solutions generalize the known…
This paper discusses the algorithms and implementations of three Mathematica packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically…
Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to…
The ``Painlev\'e analysis'' is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential…
A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved \textit{via} a method initiated by Painleve'…
The Painlev\'e test is a widely applied and quite successful technique to investigate the integrability of nonlinear ODEs and PDEs by analyzing the singularity structure of the solutions. The test is named after the French mathematician…
The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated…
We consider the non-stationary Heun equation, also known as quantum Painlev\'e VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the…
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…
This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between…