Local frequency spectra for nonlinear wave equations

For linear wave propagation one is often interested more in the local distribution of the wavevectors than in the global spectral distribution (i.e. the Fourier transform). A function that may act as such a local frequency spectrum is the real-valued Wigner distribution function. In this paper this concept of local frequency spectra is generalised to non-linear wave propagation which is governed by a class of non-linear wave equations. This class includes such well known equations as the non-linear Schrodinger equation, the Korteweg-de Vries equation and the Burgers equation. Furthermore the derivation of a transport equation for these local frequency spectra is given on the basis of the dispersion relation for the linearised wave equation. By taking local moments of this transport equation with respect to the frequency variable, an infinite hierarchy of so-called balance equations is constructed. For the non-linear Schrodinger equation the successive conservation laws (in principle, infinitely many) have been calculated straightforwardly from these balance equations.