相关论文: Movable singularities of solutions of difference e…

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…

A unique analytic continuation result is proved for solutions of a relatively general class of difference equations, using techniques of generalized Borel summability. This continuation allows for Painlev\'e property methods to be extended…

A system of nonlinear differential equations $x^{1+\gamma}\frac{dY}{dx}= F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of asymptotic expansion of transformations and solutions than preceding pioneering works, by…

The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic…

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and…

We show that there exists a rational change of coordinates of Painlev\'e's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase…

In this paper, the Painlev\'e property to fractional differential equations (FDEs) are extended and the existence and uniqueness theorems for both linear and nonlinear FDEs are established. The results contribute to the research of…

In a recent publication, it was shown that a large class of integrals over the unitary group U(n) satisfy difference equations over $n$, involving a finite number of steps; special cases are generating functions appearing in questions of…

Hyperasymptotics is an analytical method that incorporates exponentially small contributions into asymptotic approximations, thereby expanding their domain of validity, improving accuracy, and providing deeper insight into the underlying…

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using…

We study singularity confinement phenomena in examples of delay-differential Painlev\'e equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in…

We test the $\mathbb{C}P^{N-1}$ sigma models for the Painlev\'e property. While the construction of finite action solutions ensures their meromorphicity, the general case requires testing. The test is performed for the equations in the…

The Painlev\'e classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification…

We use exponential asymptotic analysis to identify the relevance of Stokes' phenomenon to integrability in discrete systems. We study Stokes' phenomenon in two discrete problems with the same (leading-order) continuous limit, a…

The paper addresses generalized Borel summability of ``$1^+$'' difference equations in ``critical time''. We show that the Borel transform $Y$ of a prototypical such equation is analytic and exponentially bounded for $\Re(p)<1$ but there is…

We study the asymptotic behaviour of the solutions of the fifth Painlev\'e equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and…

Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove…

After a brief introduction to the Painlev\'{e} property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlev\'{e} tests. The tests are…

We examine the validity of the results obtained with the singularity confinement integrability criterion in the case of discrete Painlev\'e equations. The method used is based on the requirement of non-exponential growth of the homogeneous…

Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a…