  In this report, the various 1D single soliton and multi-soliton solutions of
the Sine-Gordon equation are explored. First the topological kink solitons and
their properties for the Sine-Gordon, as well as the $\phi^{4}$ model are
investigated. This includes energy and momentum conservation laws for both
models, as well as static kinks and antikinks. The Bogomol'nyi equation and the
importance of Lorentz invariance are also discussed. Then in the second half,
the focus is on Sine-Gordon multi-kink solutions, which are found by a
separation of variables method rather than using inverse scattering or
B\"acklund transforms. These kink-kink and kink-antikink collisions are
explained by considering topological charges and interaction energies. The
special bound state dynamical breather soliton is obtained and properties such
as the binding energy of the breather are presented. Following this, the
Frenkel-Kontorova model is explored, leading to the discrete Sine-Gordon model
and the introduction of discrete breathers.
