The University of Southampton
Mathematical Sciences

# Research project: Asymptotics

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Yes

Southampton is a key centre for research in exponential asymptotics

Contrary to the impression given in many undergraduate courses, most mathematical equations cannot be solved exactly. Accurate approximations can often be derived by expanding the solution in terms of a small (or large) asymptotic parameter involved in the problem.

The problem for mathematicians is that often such asymptotic series diverge for any value of the asymptotic parameter. Poincaré gave a definition for an asymptotic expansion and so derived a calculus for their rigorous treatment. This became the generally accepted method throughout mathematics in the 20th century. The only problem is that Poincaré's definition ignores terms in the asymptotic parameter on exponentially smaller scales.

A direct consequence is that the same asymptotic Poincaré expansion represents an infinite number of functions, each separated by only exponentially small terms. Additionally, if one analytically continues the function to other values of the parameter, the asymptotic series representation is discontinuous in form as new terms suddenly appear. Such discontinuities are known as the Stokes phenomenon.

Exponential asymptotics has several roots, for example in the theory of resurgence, and theoretical physics. It exploits previously ignored universal analytic relationships between the components of asymptotic expansions to correctly incorporate the exponentially small terms that are missing in Poincaré's approach. In addition to the improvement in numerical accuaracy of expansions, the Poincaré ambiguities are removed, rigorous universal error bounds are derived and (most significantly) the range of validity of expansions in improved dramatically.

A key technique in the analysis is the study of the exponential asymptotics of a function is its Borel transform (related to an inverse-Laplace transform). This extremely powerful approach may be applied in quite general situations, ranging from linear ODEs to nonlinear PDEs, from multiple integrals to difference equations.

• The successes of exponential asymptotics include:
• the provision of universal, exact, remainder terms for whole classes of expansions,
• providing improved, tight and rigorous error bounds to asymptotic expansions,
• the hyper-exponential improvement in the numerical approximations of solutions,
• the systematic calculation of Stokes constants, the skeleton on which all asymptotic expansions are hung,
• the extraction of new geometric information from spectral eigenvalue functions,
• the study of long time-asymptotics of nonlinear PDEs,
• solvability conditions for PDE problems.

Southampton is a key centre for research in exponential asymptotics with active collaborations with other researchers in the US, UK, Europe, Asia and Australia.

Recent work at Southampton has uncovered the systematic role of a new (and so previously neglected) hierarchy of so-called "higher-order Stokes phenomena" which, although they take place at hyper-exponentially small levels, can affect the whether a Stokes phenomenon takes place with consequent leading-order changes in the asymptotic expansions, and so the behaviour of the solution.

One of the most significant affects of the higher-order Stokes phenomenon so far discovered is an explanation of an apparent (and overlooked) paradox arising in conventional asymptotics: why smoothed nonlinear shock waves are not caustics.

The research in this expanding area is continuing apace....

### Related research groups

Applied Mathematics and Theoretical Physics