Loss of a Free Energy Functional Through Boundary Conditions

Thursday, April 3, 2014 - 10:30am - 11:30am
Lind 409
David Morrissey (University of Minnesota, Twin Cities)
Pattern forming PDE often rely on a Lyapunov functional for uniqueness and existence, in turn the existence of such functionals often rely on infinitely extended domains. Exotic boundary conditions break this variational structure. I will present an overview of the Cahn-Hilliard equation and Swift-Hohenberg equation as examples of such pattern forming differential equations. For small bifurcation parameter values in the Swift-Hohenberg equation it is well known that there exists a family of solutions parameterized by the wavenumber. For the Swift-Hohenberg equation I will show a Numerical Homotopy of this family of solutions from Neumann to Transparent boundary conditions on a simulated half line which exhibit a transition from phase selection to wavenumber selection.