  This paper proposes a fairly general new point of view on the question of
asymptotic stability of (topological) solitons. Our approach is based on the
use of the distorted Fourier transform at the nonlinear level; it does not rely
on Strichartz or virial estimates and is therefore able to treat low power
nonlinearities (hence also non-localized solitons) and capture the global (in
space and time) behavior of solutions. More specifically, we consider quadratic
nonlinear Klein-Gordon equations with a potential in one space dimension. The
potential is assumed to be regular, decaying, and either generic or exceptional
(with some additional parity assumptions). Assuming that the associated
Schr\"odinger operator has no negative eigenvalues, we obtain global-in-time
bounds, including sharp pointwise decay and modified asymptotics, for small
solutions. These results have implications for the asymptotic stability of
solitons, or topological solitons, for a variety of problems. For instance, we
obtain full asymptotic stability of kinks with respect to odd perturbations for
the double Sine-Gordon problem (in an appropriate range of the deformation
parameter). For the $\phi^4$ problem, we obtain asymptotic stability of the
kink (with respect to odd perturbations) when the coupling to the internal mode
appearing in the linearization around it is neglected. Our results also go
beyond these examples since our approach allows for the presence of a fully
coherent phenomenon at the level of quadratic interactions, which creates a
degeneracy in distorted Fourier space. We devise a suitable framework that
incorporates this, and use multilinear harmonic analysis in the distorted
setting to control all nonlinear interactions.
