Dynamics of domain walls governed by the convective Cahn–Hilliard equation

The convective Cahn–Hilliard (CCH) equation, u t + ( u x x + u − u 3 ) x x − ( D / 2 ) ( u 2 ) x = 0 , has been suggested recently for the description of several physical phenomena, including spinodal decomposition of (driven) phase separating systems in an external field, instability of steps moving on a crystal surface, and faceting of growing, thermodynamically unstable surfaces. In this paper the dynamics of domain walls (kinks) governed by the convective Cahn–Hilliard equation is studied by means of asymptotic and numerical methods. A special attention is paid to the dynamics of kink pairs and triplets that play crucial role in the coarsening (Ostwald ripening) process. For the driving parameter D < D 0 = 2 / 3 and large distance between the kinks, L ≫ 1 , analytical formulas are found that describe the motion of the kink pairs and triplets. The analytical formulas are in excellent agreement with the results of direct numerical simulations of CCH equation. They explain the logarithmically slow coarsening observed for the kink dynamics governed by CCH equation when the distances between the kinks become large, unlike the fast coarsening for moderate distances between the kinks at intermediate stages.