  KEEN waves are nonlinear, non-stationary, self-organized asymptotic states in
Vlasov plasmas outside the scope or purview of linear theory constructs such as
electron plasma waves or ion acoustic waves. Nonlinear stationary mode theories
such as those leading to BGK modes also do not apply. The range in velocity
that is strongly perturbed by KEEN waves depends on the amplitude and duration
of the ponderomotive force used to drive them. Smaller amplitude drives create
highly localized structures attempting to coalesce into KEEN waves. These cases
have much more chaotic and intricate time histories than strongly driven ones.
The narrow range in which one must maintain adequate velocity resolution in the
weakly driven cases challenges xed grid numerical schemes. What is missing
there is the capability of resolving locally in velocity while maintaining a
coarse grid outside the highly perturbed region of phase space. We here report
on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative
non-uniform cubic splines in velocity that tackles this problem head on. An
additional feature of our approach is the use of a new high-order
time-splitting scheme which allows much longer simulations per computational e
ort. This is needed for low amplitude runs which take a long time to set up
KEEN waves, if they are able to do so at all. The new code's performance is
compared to uniform grid simulations and the advantages quanti ed. The birth
pains associated with KEEN waves which are weakly driven is captured in these
simulations. These techniques allow the e cient simulation of KEEN waves in
multiple dimensions which will be tackled next as well as generalizations to
Vlasov-Maxwell codes which are essential to understanding the impact of KEEN
waves in practice.
