169,798 research outputs found
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
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)
When more can be less: book review
No full textAn Introduction to Ordinary Differential Equations first edition By James C. Robinson Cambridge University Press 399pp, Pounds 65.00 and Pounds 24.99 ISBN 0 521 82650 0 and 53391 0Differential Equations with Mathematica third edition By Martha L. Abell and James P. Braselton Academic Press, 876pp, Pounds 40.95 ISBN 0 12 041562 3My seven-year-old returned from school last week waving a mathematics syllabus. Over the next three years, she will learn to multiply two-digit numbers in three different ways. If anything will confuse all but the swiftest of learners, it must surely be having to learn a new algorithm each year. This approach wastes two years during which more advanced material could be covered, and has consequences throughout the mathematical education of a UK student. I therefore sympathise with James Robinson, author of An Introduction to Ordinary Differential Equations. His stated motivation for writing the book is that he could not find a suitable text that covered less-advanced first-year undergraduate mathematics topics at length. Indeed, the first 40 per cent of the book covers material previously firmly embedded in A-level study. Differential Equations with Mathematica by Martha Abell and James Braselton also deals with contemporary first-year undergraduate mathematics courses in differential equations. Both books cover equations of first and second order, numerical solutions, coupled linear and nonlinear systems of equations. There, however, the similarity ends.Robinson's approach is straightforward but includes novel examples. When discussing resonance, he chooses not to discuss the cliched Tacoma Narrows Bridge collapse, but uses the marching-related failure of the Broughton suspension bridge in 1831. Continuing the design theme, London's Millennium Bridge features as one of the many exercises and provides the jacket illustration.The middle sections cover series solutions, numerics and difference equations. To this point, critics might comment that much of the material could be found in the plethora of texts containing "mathematics", "engineering" and "advanced" in the title. However, the style of the final two sections, covering coupled linear and nonlinear equations, with hints of what can be done later, directs the book at mathematicians.The very title of Differential Equations with Mathematica rules out its use on many courses. But the authors have ecumenically written an almost identical book involving Maple. Titles using Matlab also exist.Linking the learning of fundamental topics so firmly with computer packages runs the obvious risk of germinating mathematical couch potatoes. Courses now often involve students "working through" pre-prepared computer notebooks. Theoretically, this "innovative" solution satisfies the nebulous concept of "student-centred learning" while minimising the associated increase in staff workloads.In practice, however, some students sit, chins in hands, rhythmically tapping return keys, staring blankly at the screen. Finding that this is not the Game Boy entertainment they expect from university, they fire up their MP3 players and retreat into the Darkness.Some readers will find the repeated interruption of basic definitions and examples with code in Differential Equations with Mathematica extremely distracting. Conversely, the presence of code tempts the motivated student into active interaction. Rather than passively accepting the complexity of a dynamical system, they can easily plot the trajectories to understand the structure better. Faced with the Fourier series expansion of a discontinuous function, for example, they observe the Gibbs phenomenon experimentally. It is that type of student who will benefit most from this text.Interestingly, this book contains many examples rooted in biological systems. It deals with Fourier series, Laplace transforms and partial differential equations, and so it can be used to teach engineers. One concern is that the nomenclature of notebooks on the accompanying CD does not appear to correspond with the printed book.The contrasting approaches of these two texts indicate that undergraduate mathematics is again on the cusp. A little-noticed statement in the recent Tomlinson report suggests that the A-level syllabus will shrink by a third in the new diploma.If the recommendations of this report are implemented in full, I fear that by the time my overmultiplied daughter leaves school, Robinson will have had to write another book.Christopher Howls is senior lecturer in applied mathematics, Southampton University
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
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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
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
High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon
No full textA formalism is developed for calculating high coefficients cr of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂B. The cr are used to test the following conjectures. (a) The sequence of cr diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. (b) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B. (c) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series
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
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