Nonlinear orbital stability of traveling wave solutions to an elasto-chemical model

We address the nonlinear stability of a family of traveling wave solutions to the system proposed by Lane et al. (IMA J. Math. Appl. Med. Biol. 4 (1987), no. 4, pp. 309-331), to model a pair of mechano-chemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg’s surface, and a free calcium concentration front. These waves have been observed on the surface of some vertebrate eggs shortly after fertilization. The family of waves is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. It is shown that small perturbations of the waves yield solutions to the nonlinear equations which decay exponentially to a phase-modulated traveling wave.

This is joint work with Gilberto Flores (UNAM).