Related papers: Painlev\'e analysis for nonlinear partial differen…
We consider solutions of a discrete Painlev\'e equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation…
We derive the integrability conditions of nonautonomous nonlinear Schr$\rm\ddot o$dinger equations using the Lax Pair and Similarity Transformation methods. We present a comparative analysis of these integrability conditions with those of…
Separate consideration of properties of roots of Third Painlev\'e transcendents (P_III-functions) is necessary due to irregularity the differential equation defining them reveals on the subset of the phase space where its solution would…
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 ${\epsilon}^2$ .…
We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in…
In arXiv:1711.10132 a new approximating invariant ${\mathsf{TC}}^{\mathcal{D}}$ for topological complexity was introduced called $\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of…
Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as "symmetries" or the B\"acklund transformations in Painlev\'{e} equations. We…
We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlev\'e equation and the difference second Painlev\'e equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the…
We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a…
The integrability of a system of two symmetrically coupled higher-order nonlinear Schr\"{o}dinger equations with parameter coefficients is tested by means of the singularity analysis. It is proven that the system passes the Painlev\'{e}…
This paper is an addendum to earlier papers \cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painlev\'e I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the…
The second Painlev\'e hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlev\'e equation,…
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,…
This article deals with an inverse problem of identifying the fractional order in the 1D time fractional diffusion equation (TFDE in short) using the measurement at one space-time point. Based on the expression of the solution to the…
We study a fully noncommutative generalisation of the commutative fourth Painlev\'e equation that possesses solutions in terms of an infinite Toda system over an associative unital division ring equipped by a derivation.
Motivated by the uniqueness problem for monostable semi-wavefronts, we propose a revised version of the Diekmann and Kaper theory of a nonlinear convolution equation. Our version of the Diekmann-Kaper theory allows 1) to consider new types…
We consider the symmetric gap probability distributions of certain Freud unitary ensembles. This problem is related to the Hankel determinants generated by the Freud weights supported on the complement of a symmetric interval. By using Chen…
Although cosmological solutions to Einstein's equations are known to be generically singular, little is known about the nature of singularities in typical spacetimes. It is shown here how the operator splitting used in a particular…
We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Pad\'{e} approximants and conformal mappings, for solving quasi-analytically a…
This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…