  Excitation of large amplitude strongly nonlinear ion acoustic waves from a
trivial equilibrium by a chirped frequency drive is discussed. Under certain
conditions, after passage through the linear resonance in this system, the
nonlinearity and the variation of parameters work in tandem to preserve the
phase-locking with the driving wave via excursion of the excited ion acoustic
wave in its parameter space, yielding controlled growth of the wave amplitude.
We study these autoresonant waves via a fully nonlinear warm fluid model and
predict formation of sharply peaked (extreme) ion acoustic excitations with
local ion density significantly exceeding the unperturbed plasma density. The
driven wave amplitude is bounded by the kinetic wave-breaking, as the local
maximum fluid velocity of the wave approaches the phase velocity of the drive.
The Vlasov-Poisson simulations are used to confirm the results of the fluid
model and the Whitham's averaged variational principle is applied in analyzing
evolution of the autoresonant ion acoustic waves.
