Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow

TL;DR

This paper provides a rigorous numerical proof of the linear and nonlinear stability of small-amplitude periodic roll waves in thin film flow, using interval arithmetic and a novel bootstrap approach for efficient computation.

Contribution

It introduces a new numerical verification method combining interval arithmetic and analytic interpolation to establish stability of roll waves in the KdV-KS model.

Findings

Confirmed stability of small-amplitude roll waves in the KdV limit

Developed a two-step parameter space partitioning for rigorous evaluation

Established a framework for spectral stability analysis of similar systems

Abstract

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.