Separated spin evolution quantum hydrodynamics of degenerate electrons with spin-orbit interaction and extraordinary wave spectrum

To consider the contribution of the spin-orbit interaction in the extraordinary wave spectrum we derive a generalization of the separate spin evolution quantum hydrodynamics. Applying the corresponding nonlinear Pauli equation we include the Fermi spin current contribution in the spin evolution. We find that the spectrum of extraordinary waves consists of three branches: two of them are well-known extraordinary waves and the third one is the spin-electron acoustic wave. A change of the extraordinary wave spectrum due to the spin-orbit interaction is also obtained.