Related papers: Painleve equations, vector fields, and ranks in di…
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification,…
There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…
We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the…
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…
We present a new approach to determine the rational solutions of the higher order Painleve equations associated to periodic dressing chain systems. We obtain new sets of solutions, giving determinantal representations indexed by specific…
We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three…
Field equations for generalized principle models with nonconstant metric are derived and ansatz for their Lax pairs is given. Equations that define the Lax pairs are solved for the simplest solvable group. The solution is dependent on one…
A discrete analogue of the holomorphic map z^a is studied. It is given by a Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding immersed circle patterns lead to special separatrix solutions…
A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial…
We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an…
We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform…
Based on the so-called re-scaling method, we will give a detailed description of the solutions to the Hamiltonian system (\ref{Hsystem}) below, which was discovered only recently by Kecker, and is strongly related to Painleve's fourth…
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…
Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations…
The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated…
We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around…
In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the…
Two-component hyperbolic system of equations generated by ordinary differential Painlev\'e I \[ u_{yy}=6u^2+y \] and Painlev\'e III \[ yuu_{yy}=yu^2_{y}-uu_y+\delta y+\beta u+\alpha u^3 +\gamma yu^4 \] equations are considered, where…
A variational approach is presented to calculate the stress field generated by a system of dislocations. It is shown that in the simplest case, when the material containing the dislocations obeys Hooke's law the variational framework gives…
A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The…