From the conserved Kuramoto–Sivashinsky equation to a coalescing particles model

The conserved Kuramoto–Sivashinsky (CKS) equation, ∂ t u = − ∂ x x ( u + u x x + u x 2 ) , has recently been derived in the context of crystal growth, and it is also strictly related to a similar equation appearing, e.g., in sand-ripple dynamics. We show that this equation can be mapped into the motion of a system of particles with attractive interactions, decaying as the inverse of their distance. Particles represent vanishing regions of diverging curvature, joined by arcs of a single parabola, and coalesce upon encounter. The coalescing particles model is easier to simulate than the original CKS equation. The growing interparticle distance ℓ represents coarsening of the system, and we are able to establish firmly the scaling ℓ ̄ ( t ) ∼ t . We obtain its probability distribution function, g ( ℓ ) , numerically, and study it analytically within the hypothesis of uncorrelated intervals, finding an overestimate at large distances. Finally, we introduce a method based on coalescence waves which might be useful to gain better analytical insights into the model.