  Boson stars have attracted much attention in recent decades as simple,
self-consistent models of compact objects and also as self-gravitating
structures formed in some dark-matter scenarios. Direct detection of these
hypothetical objects through electromagnetic signatures would be unlikely
because their bosonic constituents are not expected to interact significantly
with ordinary matter and radiation. However, binary boson stars might form and
coalesce emitting a detectable gravitational wave signal which might
distinguish them from ordinary compact object binaries containing black holes
and neutron stars. We study the merger of two boson stars by numerically
evolving the fully relativistic Einstein-Klein-Gordon equations for a complex
scalar field with a solitonic potential that generates very compact boson
stars. Owing to the steep mass-radius diagram, we can study the dynamics and
gravitational radiation from unequal-mass binary boson stars with mass ratios
up to $q\approx23$ without the difficulties encountered when evolving binary
black holes with large mass ratios. Similar to the previously-studied
equal-mass case, our numerical evolutions of the merger produce either a
nonspinning boson star or a spinning black hole, depending on the initial
masses and on the binary angular momentum. We do not find any evidence of
synchronized scalar clouds forming around either the remnant spinning black
hole or around the remnant boson stars. Interestingly, in contrast to the
equal-mass case, one of the mechanisms to dissipate angular momentum is now
asymmetric, and leads to large kick velocities (up to a few $10^4\,{\rm km/s}$)
which could produce wandering remnant boson stars. We also compare the
gravitational wave signals predicted from boson star binaries with those from
black hole binaries, and comment on the detectability of the differences with
ground interferometers.
