20 research outputs found
Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis
The singularly perturbed Riccati equation is the first-order nonlinear ODE
in the complex domain where is a
small complex parameter. We prove an existence and uniqueness theorem for exact
solutions with prescribed asymptotics as in a halfplane. These
exact solutions are constructed using the Borel-Laplace method; i.e., they are
Borel summations of the formal divergent -power series solutions. As an
application, we prove existence and uniqueness of exact WKB solutions for the
complex one-dimensional Schr\"odinger equation with a rational potential.Comment: Paper has been reorganised; notation, terminology, and theorem
statements made clearer. Essential content unchange
A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem
We consider the overdamped limit of two-dimensional double well systems
perturbed by weak noise. In the weak noise limit the most probable
fluctuational path leading from either point attractor to the separatrix (the
most probable escape path, or MPEP) must terminate on the saddle between the
two wells. However, as the parameters of a symmetric double well system are
varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the
bifurcation point in parameter space, the activation kinetics of the system
become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a
system at the bifurcation point, by using the Maslov-WKB method to construct an
approximation to the quasistationary probability distribution of the system
that is valid in a boundary layer near the separatrix. The approximation is a
formal asymptotic solution of the Smoluchowski equation. Our analysis relies on
the development of a new scaling theory, which yields `critical exponents'
describing weak-noise behavior near the saddle, at the bifurcation point.Comment: LaTeX, 60 pages, 24 Postscript figures. Uses epsf macros to include
the figures. A file in `uufiles' format containing the figures is separately
available at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/figures.uu
and a Postscript version of the whole paper (figures included) is available
at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/paperF.p
Locating complex singularities of Burgers' equation using exponential asymptotics and transseries
Burgers' equation is an important mathematical model used to study gas
dynamics and traffic flow, among many other applications. Previous analysis of
solutions to Burgers' equation shows an infinite stream of simple poles born at
t = 0^+, emerging rapidly from the singularities of the initial condition, that
drive the evolution of the solution for t > 0.
We build on this work by applying exponential asymptotics and transseries
methodology to an ordinary differential equation that governs the small-time
behaviour in order to derive asymptotic descriptions of these poles and
associated zeros.
Our analysis reveals that subdominant exponentials appear in the solution
across Stokes curves; these exponentials become the same size as the leading
order terms in the asymptotic expansion along anti-Stokes curves, which is
where the poles and zeros are located. In this region of the complex plane, we
write a transseries approximation consisting of nested series expansions. By
reversing the summation order in a process known as transasymptotic summation,
we study the solution as the exponentials grow, and approximate the pole and
zero location to any required asymptotic accuracy.
We present the asymptotic methods in a systematic fashion that should be
applicable to other nonlinear differential equations.Comment: 30 pages, 6 figure
Metastability in a stochastic neural network modeled as a velocity jump Markov process
One of the major challenges in neuroscience is to determine how noise that is
present at the molecular and cellular levels affects dynamics and information
processing at the macroscopic level of synaptically coupled neuronal
populations. Often noise is incorprated into deterministic network models using
extrinsic noise sources. An alternative approach is to assume that noise arises
intrinsically as a collective population effect, which has led to a master
equation formulation of stochastic neural networks. In this paper we extend the
master equation formulation by introducing a stochastic model of neural
population dynamics in the form of a velocity jump Markov process. The latter
has the advantage of keeping track of synaptic processing as well as spiking
activity, and reduces to the neural master equation in a particular limit. The
population synaptic variables evolve according to piecewise deterministic
dynamics, which depends on population spiking activity. The latter is
characterised by a set of discrete stochastic variables evolving according to a
jump Markov process, with transition rates that depend on the synaptic
variables. We consider the particular problem of rare transitions between
metastable states of a network operating in a bistable regime in the
deterministic limit. Assuming that the synaptic dynamics is much slower than
the transitions between discrete spiking states, we use a WKB approximation and
singular perturbation theory to determine the mean first passage time to cross
the separatrix between the two metastable states. Such an analysis can also be
applied to other velocity jump Markov processes, including stochastic
voltage-gated ion channels and stochastic gene networks
Trans-Series Asymptotics of Solutions to the Degenerate Painlev\'{e} III Equation: A Case Study
A one-parameter family of trans-series asymptotics of solutions to the
Degenerate Painlev\'{e} III Equation (DP3E) are parametrised in terms of the
monodromy data of an associated two-by-two linear auxiliary problem via the
isomonodromy deformation approach: trans-series asymptotics for the associated
Hamiltonian and principal auxiliary functions and the solution of one of the
sigma-forms of the DP3E are also obtained. The actions of Lie-point symmetries
for the DP3E are derived.Comment: 102 page
Refined BPS structures and topological recursion - the Weber and Whittaker curves
We study properties of the recently established refined topological recursion
for some simple spectral curves associated to quadratic differentials. We prove
explicit formulas for the free energy and Voros coefficients of the
corresponding quantum curves, and conjecture expressions for all other (smooth)
genus zero degree two curves. The results can be written in terms of
Bridgeland's notion of refined BPS structure associated to the same initial
data, together with a quantum correction to the central charge. The
corresponding "invariants" appear to be new, but their interpretation in terms
of Donaldson-Thomas theory or QFT is not entirely clear