Modeling Stripe Formation on Growing Zebrafish Tailfins.

As zebrafish develop, black and gold stripes form across their skin due to the interactions of brightly colored pigment cells. These characteristic patterns emerge on the growing fish body, as well as on the anal and caudal fins. While wild-type stripes form parallel to a horizontal marker on the body, patterns on the tailfin gradually extend distally outward. Interestingly, several mutations lead to altered body patterns without affecting fin stripes. Through an exploratory modeling approach, our goal is to help better understand these differences between body and fin patterns. By adapting a prior agent-based model of cell interactions on the fish body, we present an in silico study of stripe development on tailfins. Our main result is a demonstration that two cell types can produce stripes on the caudal fin. We highlight several ways that bone rays, growth, and the body-fin interface may be involved in patterning, and we raise questions for future work related to pattern robustness.

State selection in the noisy stabilized Kuramoto-Sivashinsky equation.

In this work, we study the one-dimensional stabilized Kuramoto Sivashinsky equation with additive uncorrelated stochastic noise. The Eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band. This is consistent with the behavior of the phase diffusion constants of these states. Some connections to the phenomenon of state selection in driven out of equilibrium systems are made.

Superspiral structures of meandering and drifting spiral waves.

The spatiotemporal superstructure of meandering and drifting spiral waves is explained analytically. It is also demonstrated that the Hopf eigenmode that causes the transition to meandering waves is weakly exponentially localized at onset but grows exponentially slightly before onset.

Instability of multiple pulses in coupled nonlinear Schrodinger equations

An analytical technique for constructing multiple pulses is presented in this article; with it we uncover a homoclinic bifurcation through which multihumped solitary waves can be generated in systems of coupled nonlinear Schrodinger equations. A method is then developed to determine the (in)stability of multiple pulses produced by this mechanism. The analysis is applied to two models that describe optical phenomena in dispersive quadratic and Kerr media, respectively. It sheds considerable light upon the characteristics that predispose multiple pulses arising in this class of systems to be unstable.

Absolute versus convective instability of spiral waves

Absolute and convective instabilities of spirals are investigated using the continuous and the so-called absolute spectrum. It is shown that the nature of transport, induced by an absolute instability, is determined by spectral data of the asymptotic wave trains. The results are applied to core and far-field breakup of spiral waves in excitable and oscillatory media.