Resurgence and parametric asymptotics: exact results at all scales

Physical phenomena observed in nature can be modelled by a wide range of mathematical theories. From shock waves in non-linear optics, to turbulence in hydrodynamics, and gravitational ripples caused by collapsing black holes, scientists have a strong grasp on the complex equations governing these phenomena. Nevertheless, solving these equations can be arduous, often done numerically or by taking the parameters of the theory to be small, hence approximating the results as a series of terms. This is a procedure called perturbation theory, and it plays a major role in fields ranging from mathematics to engineering.

When the later terms of this series increase in magnitude, the series diverges. Despite this, summing a small number of such terms often gives a good approximation, and the series is called asymptotic. This behaviour, rather than problematic, is essential to build a complete understanding of physical observables. Hidden in it is a realm of phenomena so small that they disappear from perturbation theory (non-perturbative), but which can grow to eventually dominate our results, changing the physics.

These phenomena are widely found in mathematical descriptions of nature, but the existing methods to study them are mostly problem specific, with varying rigour and practicality. A systematic, unified framework of asymptotics is still missing. A powerful method which studies the intimate relation between asymptotics and non-perturbative phenomena is resurgence theory.

The proposed work aims to use resurgence to unify the different strands of research in asymptotics, bridging methods and disciplines, to obtain a comprehensive and practical theory of asymptotics. Blending the theoretical but powerful aspects of resurgence with practical, numerically driven theories of (exponential) asymptotics, we will be able to effectively construct full non-perturbative solutions to physical problems where only the perturbative series is known.

The research will be conducted in great extend at the host organisation, the Applied Mathematics group at the University of Southampton, in collaboration with Prof C.J. Howls, as well as members of the String Theory group. A strong collaboration with project partners A.B. Olde Daalhuis (Edinburgh), J. King (Nottingham) and R.Schiappa (Lisbon) will be vital to the success of this project. Indeed, by combining the fellow's expertise in the systematic, practical implementations of resurgence theory with the project partners' broad knowledge in the field of exponential asymptotics and its applications, this proposal presents a clear path to achieve this goal.

The physical problems that will be addressed during this project belong to a wide spectrum of research areas in mathematical sciences. Examples of these are: the formation of localised patterns in boundary value problems; the emergence of instabilities from time evolution in non-linear PDEs; asymptotic analysis of free energies of matrix models and gauge theories, and their dependence on coupling constant and rank of the gauge group; calculation of Stokes invariants for linear differential equations, from perturbative data or using integrability tools.

These problems occur in the areas of continuum mechanics, mathematical analysis, non-linear systems and mathematical analysis. The scope and potential inter-disciplinary impact of this research is evident from the universality of the features controlling asymptotic behaviour, which this project is set to address.

The proposed project focuses of the asymptotic analysis of physical problems with multi-parameter phase spaces, and the importance of general non-perturbative features known as the higher-order Stokes phenomena. This research project is rooted in the fields of mathematical analysis and mathematical physics, however the scope and practicality of both the proposed methods and applications span several disciplines, with potential interest in engineering and physics problems where multiple-scale effects are at play. The multidisciplinary nature of the project provides the evident appeal and importance of the proposed research and the resulting achievements. Consequently, the knowledge derived from this project, bridging methods and disciplines, will lead to innovation and renewed interest to the field of asymptotic theory, and its applications.

Direct applications sought during the course of this project will impact a wide range of subjects: the asymptotic the analysis of pattern formation of non-linear ODEs and PDEs appears in continuum mechanics and mathematical analysis; in non-linear systems, phenomena such as bifurcations and chaos can be addressed by asymptotic methods; the development of exponentially accurate numerical methods and associated error bounds impacts the field of numerical analysis; the numerous applications in mathematical physics range from string theory to quantum mechanics.

There is also great potential for applications and technology transfer to more applied fields beyond mathematics and physics. Examples of these are photonic crystals and metamaterials, microstructure growth, aeroacoustic jet noise and asymptotic homogenization of materials of composites. In all these fields, different methods exist to deal with asymptotic divergent series, although they are generally context specific and not systematic. Here lies the natural applicability of the aimed practical, comprehensive framework of exponential asymptotics.

Given the breadth of potential applications of the proposed research methods, the academic publications will be directed to journals of different disciplines (including Phys. Rev. Lett., Proc. Roy. Soc., SIAM). This will allow the research output to have a greater influence and impact, reaching out to the largest audience possible.

In view of this broad, multidisciplinary interest, outreach and dissemination activities will be especially well suited within the framework of the project. For the dissemination targeting early post- graduate students and established academics, the fellow will organise at least one international mini-workshop at leading mathematics centres, and another local workshop at the host organisation. The main objective here will be linking UK research on asymptotic methods in applied mathematics to the mathematical physics research being performed internationally.


Outreach events aimed at non-specialists and a wider audience will also be an important part of dissemination. The importance of the proposed research can be shown in an intriguing way, using phenomena dealing with rainbows, pattern formations and chaotic solutions, and can be of support in explaining why mathematics is present in our everyday lives. This will be achieved through the coordinated activities already set up by the host organisation, university open days, presentations at schools, and other events designed to engage the general public in the scientific research, such as the so-called science cafes.