Related papers: Open Problems for Painlev\'e Equations
Particular solutions of the Benney equations are constructed. Their properties are discussed.
An analysis of possible extension of the Painlev\'e test, to encompass the one-dimensional Vlasov equation, is performed. The extending requires a nontrivial generalization of the test. The proposed singularity analysis provides…
The Volterra lattice admits two non-Abelian analogs that preserve the integrability property. For each of them, the stationary equation for non-autonomous symmetries defines a constraint that is consistent with the lattice and leads to…
A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$…
Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.
Many statistical issues arise in the analysis of Particle Physics experiments. We give a brief introduction to Particle Physics, before describing the techniques used by Particle Physicists for dealing with statistical problems, and also…
In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential…
As a sequel to Kawakami-Nakamura-Sakai (arXiv:1209.3836), this series of papers constructs the complete degeneration scheme of four-dimensional Painlev\'e-type equations which includes the Painlev\'e-type equations associated with linear…
We consider the q-Painlev\'e equation of type $A_4^{(1)}$ (a version of q-Painlev\'e V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for…
The title says it all.
We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
We characterize Poincar\'{e} inequalities in metric spaces using rearrangement inequalities
We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness.
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…
A $\tau$ function formalism for Sakai's elliptic Painlev'e equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the…
The solutions of the discrete Painlev\'e equation I were constructed in terms of elliptic and hyperelliptic $\psi$ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference…
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…
We give description of rational solutions of polynomial-equations.
The discrete Painlev\'e property is precisely defined, and basic discretization rules to preserve it are stated. The discrete Painlev\'e test is enriched with a new method which perturbs the continuum limit and generates infinitely many…