Degenerate multi-solitons in the sine-Gordon equation

作者：Cen Julia; Correa Francisco; Fring Andreas^*

来源：Journal of Physics A-Mathematical and Theoretical, 2017, 50(43): 435201.

DOI：10.1088/1751-8121/aa8b7e

• sine-gordon equation
• Backlund transformations
• Darboux-Crum transformations
• Hirota's direct method
• Lame potentials
• time-delay

We construct various types of degenerate multi-soliton and multi-breather solutions for the sine-Gordon equation based on Backlund 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 Lame 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.

• 出版日期2017-10-27