# Solitons and degenerate complex solitons of the KdV equation

The solitons represent one of the most beautiful bridges between mathematics and physics, finding applications in a wide range of phenomena in nature. In this talk, first we will introduce the concept of a soliton solution and review some of their properties using as an example the history and development of the Korteweg de-Vries (KdV) equation, the classical example of a nonlinear integrable equation. The features of the KdV equation such as integrability and generating solution methods will be briefly discussed. Next, the concept of parity and time reversal symmetry (PT-symmetry) are introduced in order to discuss PT-symmetric deformations of the KdV equation. We finally report complex PT-symmetric multi-soliton solutions to the Korteweg de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of finite real energy. Such solutions, that differ to the usual multi-soliton ones, originate from degenerate energy limits of different methods. The structure of the asymptotic behaviour resulting from the integrability of the model together with its PT-symmetry ensure the reality of all of these charges, including in particular the mass, the momentum and the energy

### About Maths Colloquium

The Mathematics Colloquium is directed at students and academics working in the fields of pure and applied mathematics, and statistics.

We aim to present expository lectures that appeal to our wide audience.

Information for speakers

#### Information for speakers

Maths colloquia are usually held on Mondays, from 2pm to 3pm, in various locations at St Lucia.

Presentations are 50 minutes, plus five minutes for questions and discussion.

Available facilities include:

- computer
- data projector
- chalkboard or whiteboard

To avoid technical difficulties on the day, please contact us in advance of your presentation to discuss your requirements.