Investigation of one dynamic system arising in a two-fluid medium

The Cauchy problem for a one-dimensional homogeneous system of the Hopf-type equations arising in a two-fluid medium is considered. It is believed that energy dissipation occurs only due to the friction coefficient (analogous to Darcy) and the Cauchy data are given in the form of a finite trigonometric Fourier series. Recursive systems of ordinary differential equations for amplitudes are obtained.