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Degenerate multi-solitons in the sine-Gordon equation

Cen, J., Correa, F. & Fring, A. (2017). Degenerate multi-solitons in the sine-Gordon equation. Journal of Physics A: Mathematical and Theoretical, 50(43), 435201. doi: 10.1088/1751-8121/aa8b7e


We construct various types of degenerate multi-soliton and multi-breather solutions for the sine-Gordon equation based on Bäcklund transformations, Darboux–Crum transformations and Hirota's direct method. We compare the different solution procedures and study the properties of the solutions. Many of them exhibit a compound like behaviour on a small timescale, but their individual one-soliton constituents separate for large time. Exceptions are degenerate cnoidal kink solutions that we construct via inverse scattering from shifted Lamé potentials. These type of solutions have constant speed and do not display any time-delay. We analyse the asymptotic behaviour of the solutions and compute explicit analytic expressions for time-dependent displacements between the individual one-soliton constituents for any number of degeneracies. When expressed in terms of the soliton speed and spectral parameter the expression found is of the same generic form as the one formerly found for the Korteweg–de-Vries equation.

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
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
Text - Accepted Version
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