Periodic Travelling Waves in Dimer Granular Chains

We study bifurcations of periodic travelling waves in granular dimer chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter and the periodic waves with the wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic travelling waves of granular monomer chains exist

Delayed Reaction Kinetics and the Stability of Spikes in the Gierer--Meinhardt Model

A linear stability analysis of localized spike solutions to the singularly perturbed two-component Gierer--Meinhardt (GM) reaction-diffusion (RD) system with a fixed time delay $T$ in the nonlinear reaction kinetics is performed. Our analysis of this model is motivated by the computational study of Lee, Gaffney, and Monk [Bull. Math. Bio., 72 (2010), pp. 2139--2160] on the effect of gene expression time delays on spatial patterning for both the GM model and some related RD models. It is shown that the linear stability properties of such localized spike solutions are characterized by the discrete spectra of certain nonlocal eigenvalue problems (NLEP). Phase diagrams consisting of regions in parameter space where the steady-state spike solution is linearly stable are determined for various limiting forms of the GM model in both 1-dimensional and 2-dimensional domains. On the boundary of the region of stability, the spike solution is found to undergo a Hopf bifurcation. For a special range of exponents in the nonlinearities for the 1-dimensional GM model, and assuming that the time delay only occurs in the inhibitor kinetics, this Hopf bifurcation boundary is readily determined analytically. For this special range of exponents, the challenging problem of locating the discrete spectrum of the NLEP is reduced to the much simpler problem of locating the roots to a simple transcendental equation in the eigenvalue parameter. By using a hybrid analytical-numerical method, based on a parametrization of the NLEP, it is shown that qualitatively similar phase diagrams occur for general GM exponent sets and for the more biologically relevant case where the time delay occurs in both the activator and inhibitor kinetics. Overall, our results show that there is a critical value $T_{\star}$ of the delay for which the spike solution is unconditionally unstable for $T>T_{*}$, and that the parameter region where linear stability is assured is, in general, rather limited. A comparison of the theory with full numerical results computed from the RD system with delayed reaction kinetics for a particular parameter set suggests that the Hopf bifurcation can be subcritical, leading to a global breakdown of a robust spatial patterning mechanism

Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis