Exponential asymptotics for the physical sciences

This will be a course page for the Autumn 2024 Taught Centre Course

There are many interesting situations where exponentially-small phenomena, beyond-all-orders of regular perturbative expansions, dictate crucially important features of a physical problem. For example, such exponential asymptotics may govern the appearance of surface ripples in water. Or they may dictate the non-existence of solutions to a differential equation. This course will introduce you to a fascinating array of such problems arising in the physical sciences. It will be accessible to a wide range of students from different mathematical backgrounds; familiarity with asymptotic analysis is useful but not strictly necessary.

All meetings are online and commence 7 October and run for 8 weeks.

Meeting times:

• Mondays 3-4pm
• Wednesdays 3-4pm

Please feel free to email me at p.trinh@bath.ac.uk for any enquiries.

1. An introduction to the history of exponential asymptotics; initial review
2. Review of essential techniques in asymptotic analysis
3. The exponential integral and complementary error function
4. The Airy equation: from steepest descents to the Borel plane
5. The Pearcey equation and the higher-order Stokes phenomenon I
6. The Pearcey equation and the higher-order Stokes phenomenon II
7. On the universality of exponential asymptotics; methods for linear and nonlinear ODEs; dendritic crystal growth
8. Linear eigenvalue problems and the quantum harmonic oscillator
9. Water waves at low speeds I
10. Water waves at low speeds II
11. Viscous fingering I
12. Viscous fingering II
13. PDEs I
14. PDEs II