  We study the free energy of mixed $p$-spin spin glass models enriched with an
additional magnetic field given by the canonical Gaussian field associated with
a Ruelle probability cascade. We prove that this free energy converges to the
Hopf-Lax solution of a certain Hamilton-Jacobi equation. Using this result, we
give a new representation of the free energy of mixed $p$-spin models with soft
spins.
