Soliton solutions of the KdV equation revisited in the complex plane

Overview

Weakly nonlinear waves are described by dispersive pdes, such as the famous Korteweg–De Vries (KdV) equation, which is covered in MXB325. 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 engagement

• reviewing results from MXB325
• reading literature about solitons and their applications
• constructing soliton solutions and interpreting them in terms of their complex singularities
• re-examine the soliton solutions as a system of interacting particles

Research activities

• learn about solitons and their applications in physics
• revisit the KdV equation and explore various soliton solutions
• reinterpret soliton solutions in the complex plane
• all of this work will be undertaken under the supervision of Prof Scott McCue

Outcomes

• new insight into nonlinear pde models
• new techniques for studying nonlinear pdes
• building a new understanding of complexified pdes

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 elementary training in complex variable theory

Start date

1 November, 2024

End date

28 February, 2025

Location

Gardens Point campus (note the timing is negotiable)

Keywords

• nonlinear pdes
• weakly nonlinear waves
• solitons
• applied complex analysis
• complex singularities

Contact

Scott McCue, scott.mccue@qut.edu.au