A singular dispersion relation arising in a caricature of a model for morphogenesis

In 1983 Oster et al. proposed a model for morphogenesis consisting of a system of partial differential equations in which the dispersion relation for the problem linearised about the zero solution has a singularity. That is, the initial growth rate sigma of a small perturbation of wave number k from the zero solution tends to positive or negative infinity as k tends to some critical value kc from above or below respectively. We consider here as a caricature of the model a single partial differential equation with a similar dispersion relation in a bounded one-dimensional domain. The wave number, or equivalently the domain size, may be thought of as a bifurcation parameter. For the Neumann problem a phenomenon arises in which, as the domain size l increases past a critical value ln, the linear stability of the n-th mode jumps from one solution to a remote solution. That is, for l less than ln the trivial solution is unstable and a certain non-trivial solution is stable to perturbations of mode n, whereas for l greater than ln the opposite is true. For the Dirichlet or the Robin problem a linear stability change in the trivial solution occurs, but no corresponding change in any other solution has been found. The corresponding initial boundary value problems are then considered. An asymptotic analysis is performed in the weakly nonlinear limit in the particular case in which only one mode is unstable and gives an asymptotic solution for two classes of nonlinearity, one symmetric and the other asymmetric about u = 0. A development of the method of harmonic balance is then used to obtain approximate solutions in the strongly nonlinear case and when more than one mode may be unstable.