Exponential asymptotics for the physical sciences

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

This course will be live streamed on Mondays and Wednesdays at 3pm on Twitch via https://www.twitch.tv/ptrinh

The videos can be found either on Twitch or on a separate YouTube playlist access here. Note that it can take time for videos to be uploaded on YouTube but feel free to email me if there's one you need right away.

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

Please do not distribute.

1. Chapter 1: motivation
2. Chapter 2: review
3. Chapter 3: matched asymptotic expansions
4. Chapter 4: integral approximations
5. Chapter 5: the method of steepest descents
6. Chapter 6: the exponential integral
7. Chapter 7: the Airy equation
8. Chapter 8: the Pearcey equation (not covered in lectures)
9. Chapter 9: rules of exponential asymptotics
10. Chapter 10: dendritic crystal growth
11. Chapter 11: linear eigenvalue problems
12. Chapter 12: low Froude water waves
13. Chapter 13: Saffman-Taylor viscous fingering