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Level: BSc, MSci, MSc
Title: Nonlinear wave equations and the vector field method
Supervisor:

Dr Arick Shao
Research Area: Geometry and Analysis
Description:

This project deals with solving nonlinear wave equations and studying the long-time dynamics of these solutions. A particularly successful direction in the past thirty years is the so-called vector field method, which combines the analysis of these equations with elements of differential geometry and mathematical physics. In particular, the vector field method has proved to be robust, having been applied to a wide range of wave-type equations (e.g., the Yang-Mills, Dirac, and Einstein field equations in physics) and in a wide variety of settings (including both flat and curved spacetimes).
For the project, one considers various nonlinear wave equations with small initial data. The main goals would be the following:

  • Obtain solutions existing for long times or for all times;
  • Explore the long-time and asymptotic properties of these solutions (e.g., decay and dispersion).

Further details for the project will depend on the specific background of the student.
Further Reading:
  • C. Sogge, Lectures on Nonlinear Wave Equations. International Press, London, 2008.
Key Modules:

MTH5102 Calculus III

MTH5104 Convergence and Continuity

MTH5109 Geometry II: Knots and Surfaces

MTH5123 Differential Equations
Other Information:

While not formally required, any additional background in real analysis or partial differential equations would be helpful.
Current Availability: Yes