Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation

We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ordinary differential equation with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ordinary differential equation with a parameter. This paper provides an exponential asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study

Asymptotics of Some Convolutional Recurrences

Research supported by NSERC Research supported by NSERC and Canada Research Chair Program We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form an = an−1 + ∑n−d k=d f(n,k)akan−k where, very roughly speaking, f(n,k) behaves like a product of reciprocals of binomial coefficients. Some examples of such sequences from map enumerations, Airy constants, and Painlevé I equations are discussed in detail. 1 Main results There are many examples in the literature of sequences defined recursively using a convolution. It often seems difficult to determine the asymptotic behavior of such sequences. In this note we study the asymptotics of a general class of such sequences. We prove subexponential growth by using an iterative method that may be useful for other recurrences. By subexponential growth we mean that, for every constant D> 1, an = o(D n

Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

Black Hole Scattering from Monodromy

We study scattering coefficients in black hole spacetimes using analytic properties of complexified wave equations. For a concrete example, we analyze the singularities of the Teukolsky equation and relate the corresponding monodromies to scattering data. These techniques, valid in full generality, provide insights into complex-analytic properties of greybody factors and quasinormal modes. This leads to new perturbative and numerical methods which are in good agreement with previous results.Comment: 28 pages + appendices, 2 figures. For Mathematica calculation of Stokes multipliers, download "StokesNotebook" from https://sites.google.com/site/justblackholes/techy-zon

Peakompactons: Peaked compact nonlinear waves

This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. These peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg–de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg–de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. A simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solu..

Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)

Recently, Dorey and Tateo have investigated functional relations among Stokes multipliers for a Schr{\"o}dinger equation (second order differential equation) with a polynomial potential term in view of solvable models. Here we extend their studies to a restricted case of n+1-th order linear differential equations.Comment: 20 pages, some explanations improved, To appear in J. Phys.