  We present a rigorous numerical proof based on interval arithmetic
computations categorizing the linearized and nonlinear stability of periodic
viscous roll waves of the KdV-KS equation modeling weakly unstable flow of a
thin fluid film on an incline in the small-amplitude KdV limit. The argument
proceeds by verification of a stability condition derived by Bar-Nepomnyashchy
and Johnson-Noble-Rodrigues-Zumbrun involving inner products of various
elliptic functions arising through the KdV equation. One key point in the
analysis is a bootstrap argument balancing the extremely poor sup norm bounds
for these functions against the extremely good convergence properties for
analytic interpolation in order to obtain a feasible computation time. Another
is the way of handling analytic interpolation in several variables by a
two-step process carving up the parameter space into manageable pieces for
rigorous evaluation. These and other general aspects of the analysis should
serve as blueprints for more general analyses of spectral stability.
