  A multimode model of a continuously pumped atom laser is shown to be unstable
below a critical value of the scattering length. Above the critical scattering
length, the atom laser reaches a steady state, the stability of which increases
with pumping. Below this limit the laser does not reach a steady state. This
instability results from the competition between gain and loss for the excited
states of the lasing mode. It will determine a fundamental limit for the
linewidth of an atom laser beam.
