Related papers: Opening the Maslov Box for Traveling Waves in Skew…
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 [9], but essentially no results have been found in…
The FitzHugh-Nagumo equation, which was derived as a simplification of the Hodgkin-Huxley model for nerve impulse propagation, has been extensively studied as a paradigmatic activator-inhibitor system. We consider the version of this system…
Classical results from Sturm-Liouville theory establish that the Morse index of a one-dimensional Sturm-Liouville operator defined on $\mathbb{R}$ is equal to the number of its associated conjugate points. Recent advancements by Beck et…
We consider the stability of nonlinear traveling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard…
In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse…
We use the Maslov index to study the spectrum of a class of linear Hamiltonian differential operators. We provide a lower bound on the number of positive real eigenvalues, which includes a contribution to the Maslov index from a non-regular…
The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems. In this paper, we apply…
In recent work, Baird et al. have introduced a generalized Maslov index which allows oscillation techniques that have previously been restricted to eigenvalue problems with underlying Hamiltonian structure to be extended to the…
We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling…
We give a geometric proof of spectral stability of travelling kink wave solutions to the sine-Gordon equation. For a travelling kink wave solution of speed $c \neq \pm 1$, the wave is spectrally stable. The proof uses the Maslov index as a…
The aim of this paper is to prove stability of traveling waves for integro-differential equations connected with branching Markov processes. In other words, the limiting law of the left-most particle of a (time-continuous) branching Markov…
When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We are interested in developing a robust numerical algorithm to…
In the scalar Swift-Hohenberg equation with quadratic-cubic nonlinearity, it is known that symmetric pulse solutions exist for certain parameter regions. In this paper we develop a method to determine the spectral stability of these…
Patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems. A common theme in investigating such systems is to identify the intrinsic factors responsible for such self-organization. The…
Working with a general class of linear Hamiltonian systems specified on $\mathbb{R}$, we develop a framework for relating the Maslov index to the number of eigenvalues the systems have on intervals of the form $[\lambda_1, \lambda_2)$ and…
We use the Maslov index to study the eigenvalue problem arising from the linearisation about solitons in the fourth-order cubic nonlinear Schr\"odinger equation (NLSE). Our analysis is motivated by recent work by Bandara et al., in which…
We develop a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle $S^{2n-1} \subset \mathbb{C}^n$. This can be viewed as an alternative formulation of the…
For a coupled slow--fast FitzHugh--Nagumo(FHN) equation derived from a reaction-diffusion-mechanics (RDM) model, Holzer, Doelman and Kaper in 2013 studied existence and stability of the travelling pulse, which consists of two fast orbit…
We study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix…
We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique…