Diverse phenomena in conservative surface dynamics with broken parity

Local, conservative surface dynamics with linear instabilities in systems ranging from ion-bombarded surfaces to step flow in homoepitaxy to wind-blown sand have all been described by the conserved Kuramoto-Sivashinsky (cKS) equation. The cKS equation is well understood and exhibits coarsening. When parity symmetry is absent, it would seem that a nonlinear term of the form $\partial_x (\partial_x u)^2$ ought to be included, but its effects do not appear to have been studied previously. In the absence of the cKS nonlinearity the parity-breaking nonlinearity would lead to finite-time singularities; but when both nonlinear terms are present the solutions are well behaved. At least two qualitatively distinct behaviors are possible, depending on parameters: coarsening, as in the cKS equation, and non-coarsening ``traveling wave'' solutions with prominent features that are mostly frozen-in at short times and an envelope that evolves much more slowly than the wave speed.