  Nonlinear evolution of a continuous spectrum of unstable waves near the first
bifurcation point in circular Couette flow has been investigated. The
disturbance is represented by a Fourier integral over all possible axial
wavenumbers, and an integrodifferential equation for the amplitude-density
function of a continuous spectrum is derived. The equations describing the
evolution of monochromatic waves and slowly-varying wave-packets of classical
weakly nonlinear instability theories are shown to be special limiting cases.
Numerical integration of the integrodifferential equation shows that the final
equilibrium state depends on the initial disturbance, as observed
experimentally, and it is not unique. The predicted range of wavenumbers for
stable supercritical Taylor vortices is found to be narrower than the span of
the neutral curve from linear theory. Taylor-vortex flows with wavenumbers
outside this range are found to be unstable and to decay, but to excite another
wave inside the narrow band. This result is in agreement with the Eckhaus and
Benjamin-Feir sideband instability. The presence of multiple solutions at a
fixed Reynolds number for a given geometry in Taylor-Couette flows has been
known since Coles monumental contribution in 1965. It is worthwhile to note
that the existence of multiple solutions, found by Coles, differs from current
popular bifurcation theories. The multiple solutions in Coles sense have also
been found for mixed-convection flows (Yao & Ghosh Moulic 1993, 1994) besides
the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense,
which complements the bifurcation theories, is a generic property for all fluid
flows.
