  In the underdense (blowout) regime of plasma wakefield acceleration (PWFA),
the particle beam is denser than the plasma. Under these conditions, the plasma
electrons are nearly completely rarefacted from the beam channel, resulting in
a nominally uniform ion column. Extensive investigations of this interaction
assuming axisymmetry have been undertaken. However, the plasma blowout produced
by a transversely asymmetric driver possesses quite different characteristics.
They create an asymmetric plasma rarefaction region (bubble) which leads to
asymmetric focusing in the two transverse planes. This is also accompanied by
an undesired non-uniform accelerating gradient. The asymmetric blowout
cross-section is found through simulation to be elliptical, and treating it as
such permits a simple extension of the symmetric theory. In particular,
focusing fields linear in both transverse directions exist in the bubble. The
form of the wake potential and the concomitant matching conditions in this
elliptical cavity are discussed in this paper. We also discuss bubble boundary
estimation in the long driver limit and applications of the asymmetric features
of the wakefield.
