In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.

We consider a Hermitian matrix valued function A(x) ∈ C n×n , smoothly depending on parameters x ∈ Ω ⊂ R 3 , where Ω is an open bounded region of R 3 . We develop an algorithm to locate parameter values where the eigenvalues of A coalesce: conical intersections of eigenvalues. The crux of the method requires one to monitor the geometric phase matrix of a Schur decomposition of A, as A varies on the surface S bounding Ω. We develop (adaptive) techniques to find the minimum variation decomposition of A along loops covering S and show how this can be used to detect conical intersections. Further, we give implementation details of a parallelization of the technique, as well as details relative to the case of locating conical intersections for a few of A's dominant eigenvalues. Several examples illustrate the effectiveness of our technique.

We consider matrix valued functions of two parameters in a simply connected\ud region $\Omega$. We propose a new criterion to detect when such functions\ud have coalescing singular values. For {\it generic\/} coalescings,\ud the singular values come together in a ``double cone''-like intersection.\ud We relate the existence of any such singularity to the periodic structure of the\ud orthogonal factors in the singular value decomposition of the one-parameter\ud matrix function obtained restricting to closed loops in $\Omega$.\ud Our theoretical result is very amenable to approximate numerically the\ud location of the singularities

It is well known that the nontrivial solutions of the equationblow up in finite time under suitable hypotheses on the initial data, κ and f . These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher-Kolmogorov equation or Swift-Hohenberg equation.In this paper we provide details of the blow-up profile. The key idea is to relate this blow-up profile to the existence of periodic solutions for an auxiliary equation.

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