| Abstract: |
| Six-wave interactions arise in several physically relevant systems, including photon cascades in optical wave turbulence and Kelvin-wave interactions in superfluids in quantum wave turbulence. While four-wave interactions are typically dominant in systems of nonlinearly interacting waves, the resonant interactions corresponding to the dispersion relation $\omega = k^2$ are trivial in one dimension. As a result, these interactions do not appear in the statistical description of the one-dimensional system. In this setting, higher-order interactions must be considered, and the six-wave kinetic equation provides the relevant model. We initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation, which is derived from the quintic nonlinear Schrodinger equation. More precisely, we prove the existence and uniqueness of nonnegative mild solutions. We also analyze their long-time behavior, proving scattering and bijectivity of the corresponding wave operators. This is joint work with N. Pavlovic and M. Taskovic. |
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