Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of Christianson and Metcalfe [Indiana Univ. Math. J. 63 (2014), pp. 969–992] on inflection-transmission type trapping on warped product manifolds to the case of multi -warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping. [ABSTRACT FROM AUTHOR]
A unified geometric approach for the stability analysis of traveling pulse solutions for reaction diffusion equations with skew-gradient structure has been established in a previous paper (see Paul Cornwell [Indiana Univ. Math. J. 68 (2019), pp. 1801-1832]), but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of FitzHugh-Nagumo equations. [ABSTRACT FROM AUTHOR]