Pulse-train solutions of a spatially-heterogeneous amplitude equation arising in the subcritical instability of narrow-gap spherical Couette flow

We investigate some complex solutions a(x,t)$a(x,t)$ of the heterogeneous complex-Ginzburg–Landau equation

in which the real driving coefficient λ(x)$\lambda \left(x\right)$ is either constant or the quadratic $\lambda \left(0\right)-{{\rm Y}}_{\epsilon }^{2}x2$. This CGL equation arises in the weakly nonlinear theory of spherical Couette flow between two concentric spheres caused by rotating them both about a common axis with distinct angular velocities in the narrow gap (aspect ratio ε$\epsilon$) limit. Roughly square Taylor vortices are confined near the equator, where their amplitude modulation a(x,t)$a(x,t)$ varies with a suitably ‘stretched’ latitude x$x$. The value of Υ_ε${{\rm Y}}_{\epsilon }$, which depends on sphere angular velocity ratio, generally tends to zero with ε$\epsilon$. Though we report new solutions for Υ_ε≠0${{\rm Y}}_{\epsilon }\ne 0$, our main focus is the physically more interesting limit Υ_ε=0${{\rm Y}}_{\epsilon }=0$.

When