  We investigate the evolution of localized initial value profiles when
propagated in integrable versions of higher time-derivative theories. In
contrast to the standard cases in nonlinear integrable systems, where these
profiles evolve into a specific number of N-soliton solutions as dictated by
the conservation laws, in the higher time derivative theories the theoretical
prediction is that the initial profiles can settle into either two-soliton
solutions or into any number of N-soliton solutions. In the latter case this
implies that the solutions exhibit oscillations that spread in time but remain
finite. We confirm these analytical predictions by explicitly solving the
associated Cauchy problem numerically with multiple initial profiles for
various higher time-derivative versions of integrable modified Korteweg-de
Vries equations. In the case with the theoretical possibility of a decay into
two-soliton solutions, the emergence of underlying singularities may prevent
the profiles from fully developing or may be accompanied by oscillatory,
chargeless standing waves at the origin.
