arXiv subjects

Christopher J. Howls

Publications and source records attributed to Christopher J. Howls.

4 recordsLinked to original sources

arXiv2026

Which Saddles Contribute? The South-East Rule for Multidimensional Integrals

In this paper, we introduce and demonstrate a simple geometric algorithm to determine which critical points, both complex as well as real, contribute to the asymptotic evaluation of multiple integrals with exponential integrands of the form $e^{ikf(\boldsymbol{x})}$ over $\mathbb R^d$, for finite $d\ge 1$ and $f$ is analytic. In so doing, the algorithm removes the need to compute the flows of $-\text{Re} (i\nabla f)$ in $\mathbb C^d$ that is required to identify such relevant critical points in Picard-Lefschetz approaches to the derivation of such asymptotic expansions. By contrast, our algorithm relies on the combination of three simple features: the values of $f$ at all the critical points plotted in the complex Borel plane, the concept of adjacency between such points derived from algebraic resurgence/hyperasymptotic approaches and the new result here of a geometric "South-East" rule. The algorithm incorporates functions $f$ that remain bounded or unbounded on $\mathbb R^d$. We illustrate this new approach with both pedagogical and advanced examples, and draw conclusions as to its importance for resolving issues associated with Wick rotations and its implications for path integrals. This is a significant step towards a systematic way of identifying instanton contributions in real-time path integrals.

math-ph↗

arXiv2025

Appearance of the higher-order Stokes phenomenon in a discrete Airy equation

We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only encountered in solutions to third-order or higher linear homogeneous, or nonlinear, differential equations. Using steepest descent and direct series methods, we derive asymptotic solutions and the Stokes structure. Our analysis shows that discretization produces a more intricate Stokes structure, containing higher-order Stokes phenomena and infinite accumulations of Stokes and anti-Stokes curves. The latter feature is a strictly nonlinear effect in continuous differential equations. We show that this unusual behavior can be generated in a discrete equation from a linear discretization. Numerical simulations confirm the predictions, and a direct comparison with the continuous Airy equation explains how the discretization alters the Stokes structure.

math-ph↗

arXiv2020

Capturing the cascade: a transseries approach to delayed bifurcations

Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to "resum" series to obtain more efficient approximations, and have been successfully widely applied in the study of continuous linear and nonlinear, single and multidimensional problems. In particular, a method known as transasymptotic resummation can be used to describe continuous behaviour occurring on multiple scales without the need for asymptotic matching. Here we apply transasymptotic resummation to discrete systems and show that it may be used to naturally and efficiently describe discrete delayed bifurcations, or "canards", in singularly-perturbed variants of the logistic map which contain delayed period-doubling bifurcations. We use transasymptotic resummation to approximate the solutions, and describe the behaviour of the solution across the bifurcations. This approach has two significant advantages: it may be applied in systematic fashion even across multiple bifurcations, and the exponential multipliers encode information about the bifurcations that are used to explain effects seen in the solution behaviour.

math.DS↗

arXiv2017

Globally exact asymptotics for integrals with arbitrary order saddles

We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder terms lead naturally to sharper novel asymptotic bounds for truncated expansions that are a significant improvement over the previous best existing bounds for quadratic saddles derived two decades ago. We also develop a comprehensive hyperasymptotic theory, whereby the remainder terms are iteratively re-expanded about adjacent saddle points to achieve better-than-exponential accuracy. By necessity of the degeneracy, the form of the hyperasymptotic expansions are more complicated than in the case of quadratic endpoints and saddles, and require generalisations of the hyperterminants derived in those cases. However we provide efficient methods to evaluate them, and we remove all possible ambiguities in their definition. We illustrate this approach for three different examples, providing all the necessary information for the practical implementation of the method.

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