Validated continuation over large parameter ranges for equilibria of PDEs

Validated continuation was introduced in [4] as means of checking that the classical continuation method applied to a Galerkin projection of a PDE provides a locally unique equilibrium to the PDE of interest. In this paper we extend the numerical technique to include a parameter that leads to better bounds on the errors associated with the Galerkin truncation. We test this method on the Swift-Hohenberg and Allen-Cahn equations on one dimensional domains. For the first equation, we find no numerical obstructions to the validated continuation technique. This is not the case for the Allen-Cahn equation.