相关论文: On the algebraic solutions of the sixth Painlev\'e…
A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found…
The number of periodic solutions to Painlev\'e VI along a Pochhammer loop is counted exactly. It is shown that the number grows exponentially with period, where the growth rate is determined explicitly. Principal ingredients of the…
Starting with a rational solution to Painleve' VI, coming from a Riccati equation, using Okamoto's theory a four-parametric rational solution is obtained.
The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular…
In a recent work, we proposed the coupled Painlev\'e VI system with $A^{(1)}_{2n+1}$-symmetry, which is a higher order generalization of the sixth Painlev\'e equation ($P_{\rm VI}$). In this article, we present its particular solution…
We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte…
The sixth Painleve equation arises from a Drinfel'd-Sokolov hierarchy associated with the affine Lie algebra of type D_4 by similarity reduction.
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
The paper concerns asymptotic studies for the sixth Painlev\'e transcendent as independent variable tends to infinity. The primary tool is averaging and the Whitham method. Elliptic ansatz, appropriate modulation equation and asymptotics…
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution…
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…
We study the degenerate Garnier system which generalizes the fifth Painlev\'{e} equation. We present two classes of particular solutions, classical transcendental and algebraic ones. Their coalescence structure is also investigated.
In this paper, we classify the algebraic isomonodromic deformations that can be obtained through restriction to generic lines of logarithmic flat connections on the complex projective plane $\mathbb{P}^2_\mathbb{C}$ whose singular locus is…
This paper is devoted to two geometric constructions related to the isomonodromic method. We follow the Drinfeld ideas and develop them in the case of the curve $X=\mathbb{P}^1\setminus\{a_1,...,a_n\}$. Thus we generalize the results of…
We construct 2x2-matrix linear problems with a spectral parameter for the Painleve equations I-V by means of the degeneration processes from the elliptic linear problem for the Painleve VI equation. These processes supplement the known…
The critical behavior of a three real parameter class of solutions of the sixth Painlev\'e equation is computed, and parametrized in terms of monodromy data of the associated $2\times 2$ matrix linear Fuchsian system of ODE. The class may…
The D7 degeneration of the Painleve-III equation has solutions that are rational functions of $x^{1/3}$ for certain parameter values. We apply the isomonodromy method to obtain a Riemann-Hilbert representation of these solutions. We…
We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $\tau$…
Fourth - order analogue to the second Painlev\'{e} equation is studied. This equation has its origin in the modified Korteveg - de Vries equation of the fifth order when we look for its self - similar solution. All power and non - power…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…