Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model

From periodic travelling waves to travelling fronts in the spike-diffuse-spike model of dendritic waves.

In the vertebrate brain excitatory synaptic contacts typically occur on the tiny evaginations of neuron dendritic surface known as dendritic spines. There is clear evidence that spine heads are endowed with voltage-dependent excitable channels and that action potentials invade spines. Computational models are being increasingly used to gain insight into the functional significance of a spine wi...

Stability of Travelling Waves

Stability of travelling waves

An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green’s functions) for the linear ope...

Heteroclinic Travelling Waves in Convex FPU-Type Chains

We consider infinite FPU-type atomic chains with general convex potentials and study the existence of monotone fronts that are heteroclinic travelling waves connecting constant asymptotic states. Iooss showed that small amplitude fronts bifurcate from convex-concave turning points of the force. In this paper we prove that fronts exist for any asymptotic states that satisfy a certain algebraic c...

Nonexistence of Slow Heteroclinic Travelling Waves for a Bistable Hamiltonian Lattice Model

The nonexistence of heteroclinic travelling waves in an atomistic model for martensitic phase transitions is the focus of this study. The elastic energy is assumed to be piecewise quadratic, with two wells representing two stable phases. We demonstrate that there is no travelling wave joining bounded strains in the different wells of this potential for a range of wave speeds significantly lower...