相关论文: The Painlev\'e methods

A model for planar phenomena introduced by Jackiw and Pi and described by a Lagrangian including a Chern-Simons term is considered. The associated equations of motion, among which a 2+1 gauged nonlinear Schr\"odinger equation, are rewritten…

A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…

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…

Using the Painlev\'e--Kovalevskaya test, we find several new matrix generalizations of the Painlev\'e-4 equation. Some limiting transitions reduce them to known matrix Painlev\'e-2 equations.

Special polynomials associated with rational solutions of the second Painlev\'{e} equation and other members of its hierarchy are discussed. New approach, which allows one to construct each polynomial is presented. The structure of the…

In the work we use integral formulas for calculating the monodromy data for the Painlev\'e-2 equation. The perturbation theory for the auxiliary linear system is constructed and formulas for the variation of the monodromy data are obtained.…

In these lectures we discuss how the Painleve equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and…

It is known that discrete Painlev\'e equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlev\'e equations in which the symmetries become clearly visible. We know how to obtain…

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…

We present a method to obtain arbitrarily accurate solutions for conservative classical oscillators. The method that we propose here works both for small and large nonlinearities and provides simple analytical approximations. A comparison…

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…

Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.

The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions…

These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental…

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…

This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of…

Differential equations with the Painlev\'e property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order…

We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…

Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of…