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 ∂a/∂t=[λ(x)+ix−|a|2]a+∂2a/∂x2, in which the real driving coefficient λ(x)λ(x) is either constant or the quadratic λ(0)−Υε2x2. 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 εε) 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 xx. The value of ΥεΥε, which depends on sphere angular velocity ratio, generally tends to zero with εε. Though we report new solutions for Υε≠0Υε≠0, our main focus is the physically more interesting limit Υε=0Υε=0.When λ=constant, uniformly bounded solutions of our CGL equation on −∞0), solutions with pulse-separation LPS were located on a finite range Lmin(λ)≤LPS≤Lmax(λ).Here, we seek new pulse-train solutions, for which the product a(x,t)exp(−ixt) is spatially periodic on the length 2L=NLPS, N∈NN∈N. The BS-mode at small λλ has N=2N=2, and on increasing λλ it bifurcates to another symmetry-broken N=2N=2 solution. Other bifurcations to N=6N=6 were located. Solution branches with NN odd, namely 3, 5, 7, were only found after solving initial value problems. Many of the large amplitude solutions are stable. Generally, the BS-mode is preferred at moderate λλ, while that preference yields to the other symmetry-broken N=2N=2 solution at larger λλ. Quasi-periodic solutions are also common. We conclude that finite amplitude solutions, not necessarily of BS-form, are robust in the sense that they persist and do not evaporate.