Depletion-limited Kerr solitons in singly-resonant optical parametric oscillators

We analyze the impact of pump depletion in the generation of cavity solitons in a singly-resonant parametrical oscillator that includes a $\chi^{(3)}$ nonlinear section. We find an analytical expression that provides the soliton existence region using variational methods, study the efficiency of energy conversion, and compare it to a driven Kerr resonator modeled by the Lugiato-Lefever equation. At high walk-off, solitons in singly-resonant optical parametric oscillators are more efficient than those formed in a Kerr resonator driven through a linear coupler.