22 research outputs found
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 . The
precise description of the exponentially small jump in the dominant solution
approaching as is given. For the asymptotic power
expansion of the dominant solution, the coefficient asymptotics is found.Comment: 19 pages, LaTe
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
Quasi-linear Stokes phenomenon for the Painlev\'e first equation
Using the Riemann-Hilbert approach, the -function corresponding to the
solution of the first Painleve equation, , with the asymptotic
behavior as 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
Asymptotics for a special solution to the second member of the Painleve I hierarchy
We study the asymptotic behavior of a special smooth solution y(x,t) to the
second member of the Painleve I hierarchy. This solution arises in random
matrix theory and in the study of Hamiltonian perturbations of hyperbolic
equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is
known and relatively simple, but it turns out to be more subtle when x and t
tend to infinity simultaneously. We distinguish a region of algebraic
asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain
rigorous asymptotics in both regions. We also discuss two critical transitional
asymptotic regimes.Comment: 19 page
An Isomonodromy Cluster of Two Regular Singularities
We consider a linear matrix ODE with two coalescing regular
singularities. This coalescence is restricted with an isomonodromy condition
with respect to the distance between the merging singularities in a way
consistent with the ODE. In particular, a zero-distance limit for the ODE
exists. The monodromy group of the limiting ODE is calculated in terms of the
original one. This coalescing process generates a limit for the corresponding
nonlinear systems of isomonodromy deformations. In our main example the latter
limit reads as , where is the -th Painlev\'e equation. We
also discuss some general problems which arise while studying the
above-mentioned limits for the Painlev\'e equations.Comment: 44 pages, 8 figure
The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation
We establish the existence of a real solution y(x,T) with no poles on the
real line of the following fourth order analogue of the Painleve I equation,
x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the
existence part of a conjecture posed by Dubrovin. We obtain our result by
proving the solvability of an associated Riemann-Hilbert problem through the
approach of a vanishing lemma. In addition, by applying the Deift/Zhou
steepest-descent method to this Riemann-Hilbert problem, we obtain the
asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure
Random Matrix Theory and the Sixth Painlev\'e Equation
A feature of certain ensembles of random matrices is that the corresponding
measure is invariant under conjugation by unitary matrices. Study of such
ensembles realised by matrices with Gaussian entries leads to statistical
quantities related to the eigenspectrum, such as the distribution of the
largest eigenvalue, which can be expressed as multidimensional integrals or
equivalently as determinants. These distributions are well known to be
-functions for Painlev\'e systems, allowing for the former to be
characterised as the solution of certain nonlinear equations. We consider the
random matrix ensembles for which the nonlinear equation is the form
of \PVI. Known results are reviewed, as is their implication by way of series
expansions for the distributions. New results are given for the boundary
conditions in the neighbourhood of the fixed singularities at of
\PVI displayed by a generalisation of the generating function for the
distributions. The structure of these expansions is related to Jimbo's general
expansions for the -function of \PVI in the neighbourhood of its
fixed singularities, and this theory is itself put in its context of the linear
isomonodromy problem relating to \PVI.Comment: Dedicated to the centenary of the publication of the Painlev\'e VI
equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard
Fuchs in 190
Generic critical points of normal matrix ensembles
The evolution of the degenerate complex curve associated with the ensemble at
a generic critical point is related to the finite time singularities of
Laplacian Growth. It is shown that the scaling behavior at a critical point of
singular geometry is described by the first Painlev\'e
transcendent. The regularization of the curve resulting from discretization is
discussed.Comment: Based on a talk given at the conference on Random Matrices, Random
Processes and Integrable Systems, CRM Montreal, June 200
Painleve I, Coverings of the Sphere and Belyi Functions
The theory of poles of solutions of Painleve-I is equivalent to the
Nevanlinna problem of constructing a meromorphic function ramified over five
points - counting multiplicities - and without critical points. We construct
such meromorphic functions as limit of rational ones. In the case of the
tritronquee solution these rational functions are Belyi functions.Comment: 33 pages, many figures. Version 2: minor corrections and minor
changes in the bibliograph