City Research Online

Abstract

Using Hirota’s direct method and Bäcklund transformations we construct explicit complex one and two-soliton solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be PT -symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex PT -symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations. The multi-soliton solutions obtained from Hirota’s method break the PT -symmetric, whereas those obtained from Bäcklund transformations are PT -invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT -symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT -symmetrizable one-soliton solutions.

Publication Type:	Article
Additional Information:	This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1751-8113/49/36/365202
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Departments:	School of Science & Technology > Mathematics
SWORD Depositor:	Symplectic Administrator

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