Dynamics of curved domain boundaries in convection patterns

Curved domain boundaries (DB's) between locally stable convection patterns are studied near the onset of convection, within the framework of the Newell-Whitehead-Segel theory [J. Fluid Mech. 38, 279 (1969); 38, 203 (1969)]. We consider the case where there exists a Lyapunov functional. By means of asymptotic methods, the equations of motion for DB's are derived, and their solutions are obtained. It is shown that the behavior of a DB depends strongly on the difference between Lyapunov functional's densities of the coexisting patterns. In the case of a nonzero difference, the normal velocity depends on the orientation of the DB, and caustics can be produced in a finite time. In the case of zero difference, the normal velocity depends on both orientation and distortion of the DB, and the DB tends typically to straighten after a long time.