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