Degenerate multi-solitons in the sine-Gordon equation. (arXiv:1705.04749v2 [nlin.SI] UPDATED)

We construct various types of degenerate multi-soliton and multi-breather
solutions for the sine-Gordon equation based on B\"{a}cklund 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 Lam\'{e} 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.

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