Wave-mean field interaction in integrable turbulence

The kinetic equation for soliton gases in the Korteweg-de Vries (KdV) equation is recast in terms of a quasi-linear system of first-order partial differential equations for the deterministic mean field (soliton condensate) coupled to a dressed kinetic equation for the stochastic waves (soliton gas). Hydrodynamic reductions are obtained and a complete set of Riemann invariants is computed.  The mean field components are genuinely nonlinear except on a lower dimensional manifold whereas the stochastic wave components are linearly degenerate.  Certain reductions are proven to also satisfy the semi-Hamiltonian property and therefore be integrable in the sense of the generalised hodograph transform.  Some physical implications of this new hydrodynamic type system will be presented, including the existence of an induced mean field, soliton gas filtering, and statistics of the wave field.  The developed approach for the KdV equation readily generalizes to other integrable evolution equations.