Modified Kuramoto-Sivashinsky equation: stability of stationary solutions and the consequent dynamics

TL;DR

This paper investigates how adding a higher-order nonlinearity to the Kuramoto-Sivashinsky equation affects the stability of steady states and the resulting dynamics, revealing conditions for coarsening and the existence of steady solutions.

Contribution

It introduces a modified Kuramoto-Sivashinsky equation with a higher-order nonlinearity and analyzes how the symmetry of this nonlinearity influences stability and coarsening behavior.

Findings

Stability depends on the derivative of interface velocity with respect to wavevector.

Coarsening occurs only if the added nonlinearity is an odd function of H_x.

Steady-state solutions are not possible if the nonlinearity is an even function.

Abstract

We study the effect of a higher-order nonlinearity in the standard Kuramoto-Sivashinsky equation: \partial_x \tilde G(H_x). We find that the stability of steady states depends on dv/dq, the derivative of the interface velocity on the wavevector q of the steady state. If the standard nonlinearity vanishes, coarsening is possible, in principle, only if \tilde G is an odd function of H_x. In this case, the equation falls in the category of the generalized Cahn-Hilliard equation, whose dynamical behavior was recently studied by the same authors. Instead, if \tilde G is an even function of H_x, we show that steady-state solutions are not permissible.