Peter W. Bates

Invariant Manifolds of Spikes

Many singularly perturbed nonlinear elliptic equations have spike-like stationary solutions. These can be found through various methods, including Lyapunov-Schmidt schemes that, in the neighborhood of a proposed spike solution, decompose the operator equation into one that is restricted to a “normal subspace” and one in a “tangential subspace”. Here, these subspaces correspond to eigenstates of the operator, linearized at an approximate spike solution, and where “tangential” means “corresponding to eigenvalues near zero”, and “normal” means “complementary”. In this talk I will describe a more global decomposition in which the “tangential subspace” is replaced by a finite-dimensional manifold of spike-like states and this manifold is invariant with respect to the corresponding nonlinear parabolic equation and also normally hyperbolic. The stationary spike-like states lie on this manifold.