Related papers: The Elliptic Representation of the General Painlev…
A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear…
In this paper we deal with an eigenvalue problem in an interface elliptic equation. We characterize the set of principal eigenvalues as a level set of a concave and regular function. As application, we study a problem arising in population…
Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The…
The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special…
We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new…
A determinant formula for a class of algebraic solutions to Painlev\'e VI equation (P$_{\rm VI}$) is presented. This expression is regarded as a special case of the universal characters. The entries of the determinant are given by the…
The critical and asymptotic behaviors of solutions of the sixth Painlev\'e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.
We introduce $3N\times 3N$ Lax pair with spectral parameter for Calogero-Inozemtsev model. The one degree of freedom case appears to have $2\times 2$ Lax representation. We derive it from the elliptic Gaudin model via some reduction…
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The…
The classical Painlev\'e equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the…
In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{\'e} equations. Such problems can be characterized by the question as to how the…
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.
We consider the symmetric q-Painlev\'e equations derived from the birational representation of affine Weyl groups by applying the projective reduction and construct the hypergeometric solutions. Moreover, we discuss continuous limits of the…
The first, second and fourth Painlev\'{e} equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton diagrams of…
An elliptic equation of order $2m$ with general nonlocal boundary-value conditions, in a plane bounded domain $G$ with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space $W_2^m(G)$ are studied.…
We discuss the associativity or WDVV equations and demonstrate that they can be rewritten as certain functional relations between the {\it second} derivatives of a single function, similar to the dispersionless Hirota equations. The…
The Li\'enard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Li\'enard equation and some equations from the…
In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming…
We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the…
We present two explicit recursions which determine the elliptic Gromov-Witten invariants of CP^3 in terms of the rational ones, and give a table up to degree 5. Unlike the rational Gromov-Witten invariants, the coefficients are negative and…