Renormalization-group for amplitude equations in cellular pattern formation with and without conservation law

A proper version of the proto renormalization-group scheme is presented to derive amplitude equations in striped pattern formation with conserved and nonconserved order parameter. In the conserved case, the result preserves the conservation law as well as the rotational invariance of the physical system. This makes a great contrast with existing singular perturbation theories such as the multiple-scales method, in which the perturbation expansion truncated at any finite order destroys those symmetries.