Related papers: The Painlev\'e Integrability Test
We show that one of the five cases of a quadratic Hamiltonian, which were recently selected by Sokolov and Wolf who used the Kovalevskaya-Lyapunov test, fails to pass the Painleve test for integrability.
In this paper we give a review on the computational methods used to characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the singularity tracking method based on the analysis of the Fourier spectrum.…
In this note, we review the notion of Painlev\'e scheme of the sixth Painlev\'e equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${\Sigma_2}$. The key method is Painlev\'e…
We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair…
Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for…
We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M.…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of the DP1 are classified under criterion of their behavior while argument tends to infinity. The appropriate theorems of existence are proved.
Statistical models can involve implicitly defined quantities, such as solutions to nonlinear ordinary differential equations (ODEs), that unavoidably need to be numerically approximated in order to evaluate the model. The approximation…
The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work…
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…
In this article we study a coupled system of differential equations with Allen-Cahn type non-linearity. Motivated by physical phenomena one of the unknowns in the system is accompanied by a singular perturbation parameter ${\epsilon}^2$ .…
Lyapunov's indirect method is an attractive method for analyzing stability of non-linear systems since only the stability of the corresponding linearized system needs to be determined. Unfortunately, the proof for finite-dimensional systems…
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…
This work is concerned with the quantification of the epistemic uncertainties induced the discretization of partial differential equations. Following the paradigm of probabilistic numerics, we quantify this uncertainty probabilistically.…
We find all non-abelian generalizations of $\text{P}_1 - \text{P}_6$ Painlev\'e systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax…
In this note, we revisit the recent work of Diakonikolas, Gouleakis, Kane, Peebles, and Price (2021), and provide an alternative proof of their main result. Our argument does not rely on any specific property of Poisson random variables…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
This manuscript develops a novel understanding of non-polar solutions of the discrete Painlev\'e I equation (dP1). As the non-autonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a…