# Degenerate multi-solitons in the sine-Gordon equation

## TL;DR

This paper constructs and analyzes various degenerate multi-soliton solutions for the sine-Gordon equation, revealing their properties, asymptotic behavior, and explicit displacements, using multiple mathematical methods.

## Contribution

It introduces new degenerate multi-soliton solutions for the sine-Gordon equation and compares different solution methods, providing explicit expressions for their displacements.

## Key findings

- Solutions exhibit compound behavior on small timescales
- Individual solitons separate at large times
- Degenerate cnoidal kink solutions have constant speed without time-delay

## Abstract

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.