  We investigate the stability of traveling front solutions to nonlinear
diffusive-dispersive equations of Burgers type, with a primary focus on the
Korteweg-de Vries-Burgers (KdVB) equation, although our analytical findings
extend more broadly. Manipulating the temporal modulation of the translation
parameter of the front and employing the energy method, we establish
asymptotic, nonlinear, and orbital stability, provided that an auxiliary
Schr\"odinger equation possesses precisely one bound state. Notably, our result
is independent of the monotonicity of the profile and does not necessitate the
initial condition to be close to the front. We identify a sufficient condition
for stability based on a functional that characterizes the 'width' of the
traveling wave profile. Analytical verification for the KdVB equation confirms
that this sufficient condition holds for the relative dispersion parameter
within an open interval $\nu \in [-0.25,0.25]$, encompassing all monotone
profiles. Utilizing validated numerics or rigorous computation, we present
acomputer-assisted proof demonstrating that the stability condition itself
holds for parameter values within the interval [0.2533, 3.9].
