Related papers: The Painlev\'e Integrability Test
We examine whether the Painleve property is necessary for the integrability of partial differential equations (PDEs). We show that in analogy to what happens in the case of ordinary differential equations (ODEs) there exists a class of…
The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free…
A new system of coupled higher-order nonlinear Schroedinger equations is proposed which passes the Painleve test for integrability well. A Lax pair and a multi-field generalization are obtained for the new system.
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 heights of iterates of the discrete Painleve equations over number fields appear to grow no faster than polynomials while the heights of generic solutions of non-integrable discrete equations grow exponentially. This gives rise to a…
There exist many situations where an ordinary differential equation admits a movable critical singularity which the test of Kowalevski and Gambier fails to detect. Some possible reasons are: existence of negative Fuchs indices, insufficient…
In this work, we establish a connection between the extended Prelle-Singer procedure with other widely used analytical methods to identify integrable systems in the case of $n^{th}$-order nonlinear ordinary differential equations (ODEs). By…
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,…
We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV…
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 argue the integrability of the generalized KdV(GKdV) equation using the Painlev\'e test. For $d( \le 2)$ dimensional space, GKdV equation passes the Painlev\'e test but does not for $d \geq 3$ dimensional space. We also apply the…
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 Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev\'{e} test, the calculation of the integrals relies on a variety of…
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.
We examine by singularity analysis an equation derived by reduction using Lie point symmetries from the Euler--Bernoulli Beam equation which is the Painlev\'{e}--Ince Equation with additional terms. The equation possesses the same…
We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around…
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 show that the new third-order complex nonlinear wave equation, introduced recently by M\"{u}ller-Hoissen [arXiv:2202.04512], does not pass the Painlev\'{e} test for integrability. We find two reductions of this equation, one 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 new three-dimensional second-order nonlinear wave equation is introduced which passes the Painleve test for integrability and possesses KdV-type multisoliton solutions. Lax integrability of this equation remains unknown.