The spectrum of travelling wave solutions to the Sine-Gordon equation

We investigate the spectrum of the linear operator coming from the sine-Gordon equation linearized about a travelling kink-wave solution. Using various geometric techniques as well as some elementary methods from ODE theory, we nd that the point spectrum of such an operator is purely imaginary provided the wave speed c of the travelling wave is not ±1. We then compute the essential spectrum of the same operator

Travelling wave solutions in a negative nonlinear diffusion-reaction model

We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, c*, and investigate its relation to the spectral stability of the travelling wave solutions.Comment: 23 pages, 10 figure

The general quadruple point formula

Maps between manifolds Mm [arrow] Nm+l (l > 0) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6 and 7-tuple point formulas, and one corresponding to sigma2 sigma0 multisingularities