Fig. 2.

Dispersion relations in the x coordinate computed via (28) for a continuous variable $k_q$, using the parameters $a=0.1$, $b=2$, with surface diffusion parameters $d_u={10}^{-3}$, and $d_v={10}^{-1}$. In a–c we take ${\mathit{D}}_{\mathit{B}}={\mathit{D}}_{\mathit{S}}$, though in d we set ${\mathit{D}}_{\mathit{B}}=d_u{\mathit{I}}_{\mathbf{2}}$, corresponding to equal bulk diffusion between species. The solid lines correspond to $max(\mathfrak{R}(\lambda ))$ for different values of $\eta$ for the bulk–surface condition, whereas the dashed line corresponds to the single-domain classical case. For c, we anticipate there is an instability for relatively low $k_q/\pi$ and large $\eta$ due to surface–bulk interaction instabilities, as exemplified in Sect. 4.4 for $h\sim \text{ord}({\epsilon }^{1/2})$ and ${\epsilon }_{\ast }^2/(3{\epsilon }^{1/2})\ll 1$ (Color figure online)