74 research outputs found
Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem
No full textThe method of steepest descents for single dimensional Laplace-type integrals involving an asymptotic parameter k was extended by Berry & Howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non-local saddlepoints. This led to an improved asymptotic expansion (hyperasymptotics) which gave exponentially accurate numerical and analytic results, based on the topography of the saddle distribution in the single complex plane of the integrand. In this paper we generalize these results to similar well-behaved multidimensional integrands with quadratic critical points, integrated over infinite complex domains. As previously pointed out the extra complex dimensions give rise to interesting problems and phenomena. First, the conventionally defined surfaces of steepest descent are no longer unique. Second, the Stokes's phenomenon (whereby contributions from subdominant saddles enter the asymptotic representation) is of codimension one. Third, we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points. The new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder. Thus the growth of the late terms in the expansion can be identified, and a hyperasymptotic scheme implemented. We show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem, Riemann sheet structure and homology associated with the multidimensional topography of the integrand.</p
Exponentially improved asymptotics for anharmonic eigenvalues
No full textContents: Part I. Exact WKB analysis of linear differential equations: Takahiro Kawai and Yoshitsugu Takei, Introduction-Exact WKB analysis of linear differential equations; its background and prospect (3-7); Takashi Aoki, Takahiro Kawai and Yoshitsugu Takei, On a complete description of the Stokes geometry for higher order ordinary differential equations with a large parameter via integral representations (9, 11-14); Setsuro Fujiié and Thierry Ramond, Exact WKB analysis and the Langer modification with application to barrier top resonances (9, 15-31); Naofumi Honda, Microlocal Stokes phenomena for holonomic modules (9, 33-38); Tatsuya Koike, On a regular singular point in the exact WKB analysis (9-10, 39-53); Tatsuya Koike, Asymptotics of the spectrum of Heun's equation and the exact WKB analysis (10, 55-70); Frédéric Pham, Multiple turning points in exact WKB analysis (variations on a theme of Stokes) (10, 71-85); Kôichi Uchiyama, Graphical illustration of Stokes phenomenon of integrals with saddles (10, 87-95); André Voros, Exact quantization method for the polynomial 1D Schrödinger equation (10, 97-108); Part II. Hyperasymptotics and asymptotics beyond all orders: C. J. Howls, Introduction-development of exponential and hyper-asymptotics (111-118); Gabriel Álvarez, Christopher J. Howls and Harris J. Silverstone, Connection formula, hyperasymptotics, and Schrödinger eigenvalues: dispersive hyperasymptotics and the anharmonic oscillator (119, 121-134); Ovidiu Costin and Rodica D. Costin, Asymptotic structure of movable singularities of solutions of nonlinear analytic differential systems (119, 135-143); E. Delabaere and C. J. Howls, Hyperasymptotics for multidimensional Laplace integrals with boundaries (119, 145-163); J. R. King [John Robert King], Interacting Stokes lines (119, 165-178); Hideyuki Majima, A vanishing theorem in asymptotic analysis with asymptotic estimates of coefficients of "asymptotic series" in several variables (120, 179-187); A. B. Olde Daalhuis, On the Borel transform of the uniform asymptotic expansion of Bessel functions of large order (120, 189-195); Part III. Asymptotic analysis and structure of non-linear differential equations: Takahiro Kawai and Yoshitsugu Takei, Introduction (199-202); Takashi Aoki, Takahiro Kawai and Yoshitsugu Takei, Can we find a new deformation of SL_J with respect to the parameters contained in ( P_J) (203, 205-208); A. R. Its and A. A. Kapaev, The irreducibility of the second Painlevé equation and the isomonodromy method (203, 209-222); Nalini Joshi, True solutions asymptotic to formal WKB solutions of the second Painlevé equation with large parameter (203, 223-229); Takahiro Kawai, Natural boundaries revisited through differential equations, infinite order or non-linear (203-204, 231-243); Masatoshi Noumi and Yasuhiko Yamada, Affine Weyl group symmetries in Painlevé type equations (204, 245-259); Kyoichi Takano, Defining manifolds for Painlevé equations (204, 261-269); Yoshitsugu Takei, An explicit description of the connection formula for the first Painlevé equation (204, 271-296)
Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation
No full textWe 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
Incomplete Airy beams: finite-energy from a sharp spectral cutoff
No full textWe present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting “incomplete Airy beam” is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its fringes. Its propagation can be described explicitly using an incomplete Airy function, from which we derive simple expressions for the beam’s total power and mean position. Asymptotic analysis reveals a simple connection between the cutoff and the region of the beam with Airy-like behavior
Exponential asymptotics and boundary value problems: keeping both sides happy at all orders.
No full textWe introduce templates for exponential asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter ? ? 0+ and have a single boundary layer at one end of the interval. For linear equations, the template is a transseries that takes the form of a sliding ladder of exponential scales. For nonlinear equations, the transseries template is a two-dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed. An exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of ? terms in the series beyond all orders in the nonlinear case. We also demonstrate how these transseries can be resummed to generate multiple-scales-type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter. Finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansio
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Improving individual identification of wolves (Canis lupus) using the fundamental frequency and amplitude of their howls: a new survey method
No full textMany bioacoustic studies have been able to identify individual mammals from variations in the fundamental frequency (F0) of their vocalizations. Other characteristics of vocalization which encode individuality, such as amplitude, are less frequently used because of problems with background noise and recording fidelity over distance. In this thesis, I investigate whether the inclusion of amplitude variables improves the accuracy of individual howl identification in captive Eastern grey wolves (Canis lupus lycaon). I also explore whether the use of a bespoke code to extract the howl features, combined with histogram-derived principal component analysis (PCA) values, can improve current individual wolf howl identification accuracies. From a total of 89 solo howls from six captive individuals, where distances between wolf and observer were short, I achieved 95.5% (+9.0% improvement) individual identification accuracy of captive wolves using discriminant function analysis (DFA) to classify simple scalar variables of F0 and normalized amplitudes. Moreover, this accuracy was increased to 100% when using histogram-derived PCA values of F0 and amplitudes of the first harmonic
When is a Stokes line not a Stokes line?
No full textDuring the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters.We introduce the concept of a higher order Stokes phenomenon, at which a Stokes multiplier itself can change value. We show that the higher order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points, why some Stokes lines are irrelevant to a given problem and why it is indispensible to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher order Stokes phenomenon can have important effects on the large time behaviour of linear partial differential equations.Subsequently we apply these techniques to Burgers equation, a non-linear partial differential equation developed to model turbulent fluid flow. We find that the higher order Stokes phenomenon plays a major, yet very subtle role in the smoothed shock wave formation of this equation
Logarithmic catastrophes and Stokes's phenomenon in waves at horizons
No full textWaves propagating near an event horizon display interesting features
including logarithmic phase singularities and caustics. We consider an acoustic
horizon in a flowing Bose-Einstein condensate where the elementary excitations
obey the Bogoliubov dispersion relation. In the hamiltonian ray theory the
solutions undergo a broken pitchfork bifurcation near the horizon and one might
therefore expect the associated wave structure to be given by a Pearcey
function, this being the universal wave function that dresses catastrophes with
two control parameters. However, the wave function is in fact an Airy-type
function supplemented by a logarithmic phase term, a novel type of wave
catastrophe. Similar wave functions arise in aeroacoustic flows from jet
engines and also gravitational horizons if dispersion which violates Lorentz
symmetry in the UV is included. The approach we take differs from previous
authors in that we analyze the behaviour of the integral representation of the
wave function using exponential coordinates. This allows for a different
treatment of the branches that gives rise to an analysis based purely on
saddlepoint expansions, which resolve the multiple real and complex waves that
interact at the horizon and its companion caustic. We find that the horizon is
a physical manifestation of a Stokes surface, marking the place where a wave is
born, and that the horizon and the caustic do not in general coincide: the
finite spatial region between them delineates a broadened horizon.Comment: 34 pages, 12 figure
Axial and focal-plane diffraction catastrophe integrals
No full textExact expressions in terms of Bessel functions are found for some of the diffraction catastrophe integrals that decorate caustics in optics and mechanics. These are the axial and focal-plane sections of the elliptic and hyperbolic umbilic diffraction catastrophes, and symmetric elliptic and hyperbolic unfoldings of the X_9 diffraction catastrophes. These representations reveal unexpected relations between the integrals
Aeroacoustic catastrophes: upstream cusp beaming in Lilley's equation
No full textThe downstream propagation of high-frequency acoustic waves from a point source in a subsonic jet obeying Lilley's equation is well known to be organized around the so-called ‘cone of silence’, a fold catastrophe across which the amplitude may be modelled uniformly using Airy functions. Here we show that acoustic waves not only unexpectedly propagate upstream, but also are organized at constant distance from the point source around a cusp catastrophe with amplitude modelled locally by the Pearcey function. Furthermore, the cone of silence is revealed to be a cross-section of a swallowtail catastrophe. One consequence of these discoveries is that the peak acoustic field upstream is not only structurally stable but also at a similar level to the known downstream field. The fine structure of the upstream cusp is blurred out by distributions of symmetric acoustic sources, but peak upstream acoustic beaming persists when asymmetries are introduced, from either arrays of discrete point sources or perturbed continuum ring source distributions. These results may pose interesting questions for future novel jet-aircraft engine designs where asymmetric source distributions arise
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