Smoothing of the Higher‐Order Stokes Phenomenon
Year: 2025
Type: article
Abstract: ABSTRACT For over a century, the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989, Berry demonstrated it is possible to smooth this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution switching on/off taking a universal error function form. Following pioneering work of Berk, Nevins, and Roberts and the Japanese school of formally exact asymptotics, the concept of the higher‐order Stokes phenomenon was introduced, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated Borel‐plane singularities transitioning between Riemann sheets. Until now, the higher‐order Stokes phenomenon has also been treated as a discontinuous event. In this paper, we show how the higher‐order Stokes phenomenon is also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second‐order nonlinear ODE, and the telegraph equation, giving rise to a ghost‐like smooth contribution present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation of the effect of the smoothed higher‐order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.
Source: Studies in Applied Mathematics
DOI: 10.1111/sapm.70008
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,550
Awards: intertwined limits in nearly integrable quantum systems and complex classical dynamics, G2203179303, G3081709163, TITLE: APPLICABLE RESURGENCE & UNIFORM ASYMPTOTICSPURPOSE: THE SPECIAL FUNCTIONS ARE ENORMOUSLY USEFUL TOOLS FOR THE MATHEMATICAL MODELING OF PHYSICAL PHENOMENA IN ALL BRANCHES OF SCIENCE AND ENGINEERING, AND HENCE DEEP UNDERSTANDING OF THE PROPERTIES OF SUCH FUNCTIONS CAN HAVE WIDE IMPACT. ONE OF THE IMPORTANT QUESTIONS IS TO DETERMINE THE BEHAVIOR OF SUCH FUNCTIONS WHEN PARTICULAR PARAMETERS BECOME VERY LARGE, VERY SMALL, OR APPROACH CERTAIN CRITICAL VALUES; THIS IS THE GOAL OF THE FIELD OF ASYMPTOTICS. THIS PROJECT SEEKS TO DEVELOP AND APPLY NEW TOOLS FOR ASYMPTOTIC ANALYSIS OF FUNCTIONS WHICH ARE SOLUTIONS TO CERTAIN TYPES OF DIFFERENTIAL EQUATIONS. ACTIVITIES TO BE PERFORMED: THE THEORY OF ASYMPTOTICS AND HYPERASYMPTOTICS WILL BE ADVANCED ALONG SEVERAL FRONTS. THESE INCLUDE: (A) EXTENDING THE THEORY OF BOREL TRANSFORMATIONS FOR NON-LINEAR DIFFERENCE EQUATIONS, INVESTIGATING THE DISCRETE PAINLEVE-I EQUATION AS A TEST CASE, (B) DEVELOPING A DEEPER UNDERSTANDING OF RESURGENCE IN EXPONENTIAL ASYMPTOTICS FOR PARTIAL DIFFERENTIAL EQUATIONS, WHICH WILL BE USED TO OBTAIN SHARP AND COMPUTABLE ERROR BOUNDS FOR THE LARGE ZEROS OF MANY SPECIAL FUNCTIONS, AND (C) THE CONSTRUCTION OF TRANSITION-REGION-EXPANSIONS FOR SECOND-ORDER LINEAR ODES WITH A SEVERAL TYPES OF TURNING POINTS.EXPECTED OUTCOMES: THE FUNDAMENTAL RESEARCH SUPPORTED BY THIS GRANT WILL CONTRIBUTE IMPORTANT NEW TECHNIQUES AND TOOLS FOR THE MATHEMATICAL ANALYSIS OF SPECIAL FUNCTIONS. IN PARTICULAR, THE ASYMPTOTIC ANALYSES DEVELOPED WILL MAKE THE EVALUATION OF SPECIAL FUNCTIONS MUCH EASIER TO ACCOMPLISH NEAR CRITICAL POINTS. THE RESEARCH SUPPORTED BY THIS GRANT WILL BE PRESENTED IN OPEN SCIENTIFIC FORUMS AND PUBLISHED IN THE OPEN SCIENTIFIC LITERATURE.INTENDED BENEFICIARIES: THE RESULTS OF THIS PROJECT SHOULD BE OF INTEREST TO SCIENTISTS AND ENGINEERS IN MANY FIELDS WHO UTILIZE SPECIAL FUNCTIONS IN THEIR THEORETICAL AND COMPUTATIONAL ANALYSES. IT SHOULD BE OF PARTICULAR INTEREST TO THOSE WHO MUST DEVELOP COMPUTER PROGRAMS TO RELIABLY EVALUATE SPECIAL FUNCTIONS. FINALLY, IT WILL BENEFIT THE NIST DIGITAL LIBRARY OF MATHEMATICAL FUNCTIONS (DLMF) PROJECT, WHICH SEEKS TO CATALOG AND DISSEMINATE INFORMATION ON THE PROPERTIES OF THE SPECIAL FUNCTIONS OF APPLIED MATHEMATICS FOR USE BY THE RESEARCH COMMUNITY. SUBRECIPIENT ACTIVITIES: THERE ARE NO PLANNED SUBAWARDS., G3223208087, G3341554066, G3691759950, G4425005604, Isaac Newton Institute for Mathematical Sciences
OpenAlex ID: W4407356671