Related papers: The Painlev\'e approach to nonlinear ordinary diff…

Problem of asymptotic description for global solutions to the six Painleve equations was investigated. Elliptic anzatzes and appropriate modulation equations were written out.

This paper concerns the discrete version of the Painlev\'e identification problem, i.e., how to recognize a certain recurrence relation as a discrete Painlev\'e equation. Often some clues can be seen from the setting of the problem, e.g.,…

A class of discrete equations is considered from three perspectives corresponding to three measures of the complexity of solutions: the (hyper-) order of meromorphic solutions in the sense of Nevanlinna, the degree growth of iterates over a…

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.…

Nonlinear Sobolev-Burgers PDEs are considered. Their solutions are investigated. A technique of noncommutative line integration is utilized for their description. A new method of PDEs solution with the help of Cayley-Dickson algebras is…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

Eigenvalue problems for linear differential equations, such as time-independent Schr\"odinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of…

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…

A multidomain spectral approach for Painlev\'e transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and…

The integrability of a system of two symmetrically coupled higher-order nonlinear Schr\"{o}dinger equations with parameter coefficients is tested by means of the singularity analysis. It is proven that the system passes the Painlev\'{e}…

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…

In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the…

We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…

We apply the Painleve test for integrability to a new discrete (differential-difference) nonlinear Schrodinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain…

We provide a simple approach for the evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind the method is to reduce the inverse transform to the solution of an ordinary…

These lecture notes provide an introduction to the theory and application of symmetry methods for ordinary differential equations, building on minimal prerequisites. Their primary purpose is to enable a quick and self-contained approach for…

The intrinsic nature of a problem usually suggests a first suitable method to deal with it. Unfortunately, the apparent ease of application of these initial approaches may make their possible flaws seem to be inherent to the problem and…

Novel sequences of approximants to solutions of Painlev\'e II on finite intervals of the real line, with Neumann boundary conditions, are constructed. Numerical experiments strongly suggest convergence of these sequences in a surprisingly…

We present a systematic method for the construction of discrete Painlev\'e equations. The method, dubbed `restoration', allows one to obtain all discrete Painlev\'e equations that share a common autonomous limit, up to homographic…

We study the analytic properties of a matrix discrete system introduced in [7]. The singularity confinement for this system is shown to hold generically, i.e. in the whole space of parameters except possibly for algebraic subvarieties. This…