  In this thesis, we first review the linearized soliton perturbation theory
developed in recent years, which is particularly simple in the one-kink sector.
Using it, the amplitude and probability of kink-meson inelastic scattering can
be simplified into a perturbative problem in the kink frame. Although the
Sine-Gordon soliton and $\phi^4$ kink, which people are usually interested in,
are both reflectionless kinks, in order to consider more general cases, we
study quantum reflective kinks and find that the amplitude of meson wave
packets at different positions during propagation corresponds to the reflection
and transmission coefficients of particles scattered by symmetric potential
barriers or potential wells in quantum mechanics. We calculate the reduced
inner product of the kink states to solve the infrared divergence problem of
non-normalizable states. Then we consider the inelastic scattering of a meson
off of a kink in a (1+1)-dimensional scalar quantum field theory. At leading
order there are three inelastic scattering processes: (1) meson multiplication
(the final state is two mesons and a kink); (2) Stokes scattering (the final
state is a meson and an excited kink); (3) anti-Stokes scattering (the initial
kink is excited and the final kink is de-excited). For the first time, we
calculate the leading-order probabilities of these three processes and the
differential probabilities for final-state mesons with different momenta. We
first obtain general results for arbitrary scalar kinks and then apply them to
the kinks of the $\phi^4$ double well model to obtain analytical and numerical
results. Finally, we believe that our method can be generalized to higher
dimensions, such as the case of monopoles.
