Wave kinetic equation in a nonstationary and inhomogeneous medium with a weak quadratic nonlinearity

We present a systematic derivation of the wave kinetic equation describing the dynamics of a statistically inhomogeneous incoherent wave field in a medium with a weak quadratic nonlinearity. The medium can be nonstationary and inhomogeneous. Primarily based on the Weyl phase-space representation, our derivation assumes the standard geometrical-optics ordering and the quasinormal approximation for the statistical closure. The resulting wave kinetic equation simultaneously captures the effects of the medium inhomogeneity (both in time and space) and of the nonlinear wave scattering. This general formalism can serve as a stepping stone for future studies of weak wave turbulence interacting with mean fields in nonstationary and inhomogeneous media.