Conclusions

We have studied the dynamical behavior of the linearized spatially inhomogeneous Hamiltonian wave systems. The canonical transformation from the Fourier variables to the new spatially-dependent variables is found for the general class of the quadratic Hamiltonians. In the new variables, the linearized dynamics is governed by the canonical diagonal Hamiltonian with the spatially-dependent dispersion relation. The waveaction transport equation which coresponds to this Hamiltonian has form (6) which is typical to WKB formalism. It was previously obtained for some specific examples, e.g. in plasmas [27] and geophysical waves [26,16]. In this paper, we have given several representative examples illustrating the general results, such as the Nonlinear Schrödinger Equation without and with condensate and a advective-type system. Further possible areas of application of this formalism include water waves on lakes with variable depth or/and presence of variable mean flow, internal waves in media with variable stratification, plasma waves on profiles with variable density, geophysical waves in media with variable background rotation rates, etc.

The new Hamiltonian formalism that is presented in this paper should be crucial for extending the WT theory to the spatially inhomogeneous systems. In the spatially homogeneous systems, quadratic term in the Hamiltonian corresponds to the first term in Eq. (70). Effect of space inhomogeneity leads to the appearance of the derivative terms in the Hamiltonian, which correspond to the slow dynamics along the rays in the $(\bf {k,x})$ -space. This effect will lead to an interesting interplay of inhomogeneity and non-linearity in wave turbulence systems. More specifically, linear dispersion relation becomes spatially dependent. Consequently, the resonance conditions change as waves propagate through inhomogeneous environment. As a result of this, waves will remain in resonance for a limited amount of time, or the members of the resonant triads will change from position to position. The physical implication of this effect may be the weakened flux of energy or other conserved quantities through the wavenumber space. Other effect may be an effective broadening of the resonances, as resonances will be altered from place to place, so the wavepacket propagating through the inhomogeneous environment will be affected by averaged dispersion relation. Another potentially interesting effect is an effective three-wave interactions in a four-wave weak turbulence systems, where the role of fourth wave is played by inhomogeneity.

In order to develop a Wave Turbulence theory for spatially inhomogeneous systems, the kinetic equation has to be obtained for the cases with such finite-time wave resonances. This is an exciting task for the future work.