Related papers: Open Problems for Painlev\'e Equations
The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions of integrable matrix evolution equations, which makes it possible to…
We consider a modified Boussinesq type equation. The Painlev\'{e} test of the WTC method is performed for this equation and it shows that the equation has weak Painlev\'{e} property. Some exact solutions are constructed.
We give explicit cubic equations for the wild character varieties corresponding to the rank 3 representations of Painlev\'e equations, and compare them to the ones of their classical rank 2 representations.
Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according…
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…
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
We will study special solutions of the fourth, fifth and sixth Painlev\'e equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura's classical solutions and…
In this article we will obtain real and complex solutions to the Painleve IV equation through supersymmetric quantum mechanics. Then we will classify them into real solution hierarchies and also the complex solution hierarchies, which are…
We offer elementary proofs for fundamental properties of solutions to the homogeneous second Painlev\'e equation.
In this paper we will study the equivalence between super-Poincar\'e inequality and some log-Sobolev type inequalities, including weak log-Sobolev inequality and super log-Sobolev inequality. The explicit relations between associated rate…
We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold…
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…
In this paper, we construct a new relation between ABS equations and Painlev\'e equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painlev\'e equations.
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
We study the analytic properties and the critical behavior of the elliptic representation of solutions of the Painlev\'e 6 equation. We solve the connection problem for elliptic representation in the generic case and in a non-generic case…
This article presents an equivalent formulation of the implicit complementarity problem. We demonstrate that solution of the equivalent formulation is equivalent to the solution of the implicit complementarity problem. Moreover, we provide…
This paper presents an overview of the current state of knowledge in the field of equivariant map algebras and discusses some open problems in this area.
We present a determinant expression for a family of classical transcendental solutions of the Painlev\'e V and the Painlev\'e VI equation. Degeneration of these solutions along the process of coalescence for the Painlev\'e equations is…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
In this work there are considered model problems for two nonlinear equations, which type depends on the solution. One of the equations may be called a nonlinear analog of the Lavrent'ev-Bitsadze equation.