Steady State Bifurcations for Phase Field Crystal Equations with underlying two Dimensional Kernel

Arnaud Rougirel (1) , Appolinaire Abourou Ella (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

This paper is concerned with the study of some properties of stationary solutions to Phase Field Crystal Equations bifurcating from a trivial solution. It is assumed that at this trivial solution, the kernel of the underlying linearized operator has dimension two. By means of the multiparameter method, we give a second order approximation of these bifurcating solutions and analyse their stability properties. The main result states that the stability of these solutions can be described by the variation of a certain angle in a two dimensional parameter space. The behaviour of the parameter curve is also investigated.

Mots clés: Phase Field Crystal Equation; Bifurcation Theory; Two Dimensional Kernel; Higher Order Elliptic Equations; Stability

Domaines: Equations aux dérivées partielles [math.AP]

Arnaud Rougirel, Appolinaire Abourou Ella. Steady State Bifurcations for Phase Field Crystal Equations with underlying two Dimensional Kernel. 2013. ⟨hal-00911041⟩