Pulse-train solutions of a spatially heterogeneous amplitude equation arising in the subcritical instability of narrow gap spherical Couette flow (Q882325)

We investigate some complex solutions \(a(x,t)\) of the heterogeneous complex-Ginzburg-Landau equation \[ \partial a/\partial t=[\lambda(x)+ix-| a| ^ 2]a+\partial^ 2a/\partial x^ 2, \] in which the real driving coefficient \(\lambda(x)\) is either constant or the quadratic \(\lambda(0)-\Upsilon^ 2_ \varepsilon x^ 2\). 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 \(\varepsilon\)) limit. Roughly square Taylor vortices are confined near the equator, where their amplitude modulation \(a(x,t)\) varies with a suitably `stretched' latitude \(x\). The value of \(\Upsilon_\varepsilon\), which depends on sphere angular velocity ratio, generally tends to zero with \(\varepsilon\). Though we report new solutions for \(\Upsilon_\varepsilon\neq 0\), our main focus is the physically more interesting limit \(\Upsilon_\varepsilon=0\). When \(\lambda =\text{const}\), uniformly bounded solutions of our CGL equation on \(-\infty<x<\infty\) have some remarkable related features, which occur at all values of \(\lambda\). Firstly, the linearised equation has no non-trivial neutral modes \(\overline a(x)\exp(i\Omega t)\) with any real frequency \(\Omega\) including zero. Secondly, all evidence indicates that there are no steady solutions \(\overline a(x)\) of the nonlinear equation either. Nevertheless, \textit{A. P. Bassom} and \textit{A.M. Soward} [J. Fluid Mech. 499, 277--314 (2004; Zbl 1163.76364)] identified oscillatory finite amplitude solutions, \[ a(x,t)=\sum_ {n\in\mathbb Z}\overline a(x-x_ n)\exp\{i[(2n+1)\Omega t+\phi_ n]\}, \] expressed in terms of the single complex amplitude \(\overline a(x)\), which is localised as a pulse on the length scale \(L_{\text{PS}}=2\Omega\) about \(x=0\). Each pulse-amplitude \(\overline a(x-x_ n)\exp(i\phi_ n)\) is identical up to the phase \(\phi_n=(-1)^n\pi/4\), is centred at \(x_ n=(n+\frac 12)L_{\text{PS}}\) and oscillates at frequency \((2n+1)\Omega\). The survival of the pulse-train depends upon the nonlinear mutual interaction of close neighbours; self-interaction is inadequate, as the absence of steady solutions shows. For given constant values of \(\lambda\) in excess of some threshold \(\lambda_{\text{MIN}}(>0)\), solutions with pulse-separation \(L_{\text{PS}}\) were located on a finite range \(L_ {\min}(\lambda)\leq L_{\text{PS}}\leq L_ {\max}(\lambda)\). Here, we seek new pulse-train solutions, for which the product \(a(x,t)\exp(-ixt)\) is spatially periodic on the length \(2L=NL_{\text{PS}}\), \(N\in\mathbb N\). The BS-mode at small \(\lambda\) has \(N=2\), and on increasing \(\lambda\) it bifurcates to another symmetry-broken \(N=2\) solution. Other bifurcations to \(N=6\) were located. Solution branches with \(N\) 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 \(\lambda\), while that preference yields to the other symmetry-broken \(N=2\) solution at larger \(\lambda\). 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.