Complex solitons, reality and degenerate structure

Julia Cen, City, University of London.

Usually, one only thinks of real soliton solutions to be physically interesting, but we found some particular complex soliton solutions to also be meaningful, due to the fact that they have real energies. We show how to construct complex multi-soliton solutions to the complex Korteweg de-Vries and sine-Gordon equations [1] using Hirota’s method, Bäcklund and Darboux-Crum transformations.

Through computing the time-delays from multi-soliton scattering, we can explain how PT-symmetry along with integrability ensures reality of energy for complex multi-solitons [2]. We also discovered how to obtain degenerate multi-solitons with these methods [3,4] and found they have time-dependent displacement limit from multi-soliton scattering rather than the usual constant displacement limit as for the non-degenerate case.

We will also briefly introduce some new nonlocal integrable systems we recently found [5].

Based on:
[1] J. Cen and A. Fring, Complex solitons with real energies, J. Phys. A 49(36), 365202 (2016)
[2] J. Cen, F. Correa and A. Fring, Time-delay and reality conditions for complex solitons, J. Math. Phys. 58(3), 032901 (2017)
[3] F. Correa and A. Fring, Regularized degenerate multi-solitons, J. High Energy Phys. 2016(9), 8 (2016)
[4] J. Cen, F. Correa and A. Fring, Degenerate multi-solitons in the sine-Gordon equation, J. Phys. A 50(43), 435201 (2017)
[5] J. Cen, F. Correa and A. Fring, Integrable nonlocal Hirota equations, arXiv:1710.11560 (2017)