  We present an approach for achieving large Kerr $\chi^{(3)}$--mediated
thermal energy transfer at the nanoscale that exploits a general coupled-mode
description of triply resonant, four-wave mixing processes. We analyze the
efficiency of thermal upconversion and energy transfer from mid- to
near-infrared wavelengths in planar geometries involving two slabs supporting
far-apart surface plasmon polaritons and separated by a nonlinear $\chi^{(3)}$
medium that is irradiated by externally incident light. We study multiple
geometric and material configurations and different classes of interveening
mediums---either bulk or nanostructured lattices of nanoparticles embedded in
nonlinear materials---designed to resonantly enhance the interaction of the
incident light with thermal slab resonances. We find that even when the entire
system is in thermodynamic equilibrium (at room temperature) and under typical
drive intensities $\sim\mathrm{W}/\mu\mathrm{m}^2$, the resulting upconversion
rates can approach and even exceed thermal flux rates achieved in typical
symmetric and non-equilibrium configurations of vacuum-separated slabs. The
proposed nonlinear scheme could potentially be exploited to achieve thermal
cooling and refrigeration at the nanoscale, and to actively control heat
transfer between materials with dramatically different resonant responses.
