  Resonant upconversion through a sixth order relativistic nonlinearity
resulting in a unique resonance was recently proposed [V. M. Malkin and N. J.
Fisch, Physical Review E 108, 045208 (2023)]. The high order resonance is a
unique non-integer multiple of a driving pump frequency resulting in a
frequency upshift by a factor of $\approx 3.73$. We demonstrate the presence,
unique requirements, and growth of this mode numerically. Through tuning waves
to high amplitude, in a mildly underdense plasma, the six-photon process may
grow more than other non-resonant, but lower order processes. The growth of the
high frequency mode remains below the nonlinear growth regime. However,
extending current numerical results to more strongly coupled resonances with
longer pulse propagation distances suggests a pathway to significant
upconversion.
