Related papers: The Painlev\'e Integrability Test

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation,…

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.,…

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…

We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…

We apply various conventional tests of integrability to the supersymmetric nonlinear Schr\"odinger equation. We find that a matrix Lax pair exists and that the system has the Painlev\'e property only for a particular choice of the free…

In the Painleve analysis of nonintegrable partial differential equations one obtains differential constraints describing the movable singularity manifold. We show, for a class of n-dimensional wave equations, that these constraints have a…

This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable…

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect…

Symmetry properties are at the basis of integrability. In recent years, it appeared that so called "twisted symmetries" are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we…

We consider meromorphic particular solutions of nonlinear ordinary differential equations and perform a perturbation {\it \`a la} Poincar\'e making their linearized equation non-Fuchsian at the movable pole and Fuchsian at infinity. When…

A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…

In this paper, we examine a generalized magma equation for rational values of two parameters, $m$ and $n$. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlev\'e ODE test is…

We investigate the Painleve analysis for a (2+1) dimensional Camassa-Holm equation. Our results show that it admits only weak Painleve expansions. This then confirms the limitations of the Painleve test as a test for complete integrability…

Penetration Testing is a methodology for assessing network security, by generating and executing possible hacking attacks. Doing so automatically allows for regular and systematic testing. A key question is how to generate the attacks. This…

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…

The integrability nature of a nonparaxial nonlinear Schr\"odinger (NNLS) equation, describing the propagation of ultra-broad nonparaxial beams in a planar optical waveguide, is studied by employing the Painlev\'e singularity structure…

Discrete Painlev\'e equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlev\'e equation as a birational map between…

Discrete dynamical systems over finite fields are investigated and their integrability is discussed. In particular, the discrete Painlev\'{e} equations and the discrete KdV equation are defined over finite fields and their special solutions…

While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle,…

Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…