Periodic Travelling Wave Selection by Dirichlet Boundary Conditions in Oscillatory Reaction-Diffusion Systems

TLDR: Numerical results suggest that when this reaction-diffusion system of the generic $\lambda$-$\omega$ form is solved on a semi-infinite domain subject to Dirichlet boundary conditions in which the variables are fixed at zero, periodic travelling waves develop in the domain.

Abstract: Periodic travelling waves are a fundamental solution form in oscillatory reaction-diffusion equations. Here I discuss the generation of periodic travelling waves in a reaction-diffusion system of the generic $\lambda$-$\omega$ form. I present numerical results suggesting that when this system is solved on a semi-infinite domain subject to Dirichlet boundary conditions in which the variables are fixed at zero, periodic travelling waves develop in the domain. The amplitude and speed of these waves are independent of the initial conditions, which I generate randomly in numerical simulations. Using a combination of numerical and analytical methods, I investigate the mechanism of periodic travelling wave selection. By looking for an appropriate similarity solution, I reduce the problem to an ODE system. Using this, I derive a formula for the selected speed and amplitude as a function of parameters. Finally, I discuss applications of this work to ecology.