Abstract
An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by the non-zero reflection coefficient of the corresponding Dirac operator. Explicit formulas for the phases are also obtained. The results demonstrate an interesting phenomenon of splitting of non-decreasing solutions into an infinite series of asymptotic solitons.
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Kirsch, W., Kotlyarov, V. Soliton Asymptotics of Solutions of the Sine-Gordon Equation. Mathematical Physics, Analysis and Geometry 2, 25–51 (1999). https://doi.org/10.1023/A:1009870508971
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DOI: https://doi.org/10.1023/A:1009870508971