  In the early universe, strong large scale random flows prevail, and when
these flows collide it produces sheets and filaments for the conventional
particle dark matter model. However, for wave dark matter, the sheets and
filaments are parallelly modulated by fine-scale and large-amplitude
interference fringes. These long fine fringes tend to be unstable initially and
subsequently deform and collapse to form halos of various sizes, perhaps down
to proto-globular cluster scales. This work aims to investigate the instability
of such fringes. Two colliding streams of quantum waves result in a standing
density wave. By contrast, two colliding beams in classical collisionless fluid
systems often give rise to beam instability, characterized by unstable
oscillations with a complex frequency. Will the quantum standing density wave
be subject to a similar instability? In fact, such unstable oscillations have
never been observed in conventional quantum mechanical systems. This study
discovers that periodic fringes can indeed undergo unstable oscillations or
beam instability. The presence of any local density null in the background
density that fails the Madelung formulation is shown to be the necessary
condition for unstable oscillations. We establish a phase diagram to
distinguish between beam instability, Jeans instability, and stable oscillation
through theoretical calculations and computer simulations. Roughly speaking,
Jeans instability almost always occurs for long-wave density perturbations;
beam instability on the other hand always occurs when the density wave is
aligned with the beam direction and moreover the period of the density wave is
twice the period of background density; lastly, both instabilities diminish for
a very low background density. We have engaged in Blocks theorem extensively in
the analysis.
