Study level

  • PhD
  • Master of Philosophy
  • Honours

Faculty/School

Faculty of Science

School of Mathematical Sciences

Topic status

We're looking for students to study this topic.

Research centre

Supervisors

Professor Scott McCue
Position
Professor
Division / Faculty
Faculty of Science

Overview

Weakly nonlinear waves are described by dispersive pdes, such as the famous Korteweg–De Vries (KdV) equation. These models have applications to a variety of phenomena in physics, including the propagation of water waves, but they are also interesting from a mathematical perspective because they can have special properties.

While the KdV equation and its variants are well-studied in the literature, a new approach is to attempt to learn about wave propagation by investigating solution behaviour in complex plane. For example, there are deep connections between complex-plane singularities and the solutions on the real line that are not well understood.

Research activities

  • Simulate the KdV equation or related equation on the real line using a variety of initial conditions.
  • Investigate the role of travelling wave solutions and solitons for the governing equation.
  • Investigate the role that similarity solutions play for both small- and large-time limits.
  • Explore links with the so-called Painlevé equations, both numerically and analytically.
  • Apply numerical algorithms to analytically continue numerical solutions into the complex plane.
  • Use matched asymptotic expansions to describe the trajectory of complex-plane singularities.
  • Relate behaviour in the complex plane to real solutions on the real line.
  • Extend the above activities to three-dimensional waves via the Kadomtsev–Petviashvili equation or related equations.

Outcomes

  • New insight into weakly nonlinear wave phenomena.
  • New techniques for studying nonlinear pdes.
  • New results for 3D waves.

Skills and experience

  • Preferably a strong background in MXB322 Partial Differential Equations, MXB325 Modelling with Differential Equations 2 and MXB326 Computational Methods 2 (or equivalent third-year undergraduate courses).
  • Preferably some training in complex variable theory.

Keywords

Contact

Contact the supervisor for more information (scott.mccue@qut.edu.au).