Monodromy approach to the scaling limits in the isomonodromy systems

The isomonodromy deformation method is applied to the scaling limits in the linear NxN matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves which describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated to the n-large asymptotics in the sequence of the bi-orthogonal polynomials with cubic potentials.Comment: Latex, 15 pages, 1 figure; submitted to the proceedings of the conference NEEDS 2002; in compare to the original version, there are minor changes in the references and in the main body of the articl

Quasi-linear Stokes phenomenon for the second Painlev\'e transcendent

Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlev\'e equation $y_{xx}=2y^3+xy-\alpha$. The precise description of the exponentially small jump in the dominant solution approaching $\alpha/x$ as $|x|\to\infty$ is given. For the asymptotic power expansion of the dominant solution, the coefficient asymptotics is found.Comment: 19 pages, LaTe

Quasi-linear Stokes phenomenon for the Painlev\'e first equation

Using the Riemann-Hilbert approach, the $\Psi$-function corresponding to the solution of the first Painleve equation, $y_{xx}=6y^2+x$, with the asymptotic behavior $y\sim\pm\sqrt{-x/6}$ as $|x|\to\infty$ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in the power-like expansion to the latter are found.Comment: version accepted for publicatio

On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev\'e equation

Using a simple operator-norm estimate we show that the solution to the second Painlev\'e equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is illustrated with several numerical examples.Comment: 8 pages, to appear in Nonlinearit