  The spin-electron-acoustic waves (sometimes called the spin-plasmons) can be
found in degenerate electron gas if the spin-up electrons and spin down
electrons move relatively each other. Here, we suggest relativistic
hydrodynamics with the separate spin evolution which allows us to study linear
and nonlinear spin-electron-acoustic waves, including the
spin-electron-acoustic solitons. Presented hydrodynamic model is the
corresponding generalization of the relativistic hydrodynamic model with the
average reverse gamma factor evolution which consists of the equations for
evolution of the following functions the partial concentrations (for spin-up
electrons and spin down electrons), the partial velocity fields, the partial
average reverse relativistic gamma factors, and the partial flux of the reverse
relativistic gamma factors. We find that the relativistic effects decreases the
phase velocity of spin-electron-acoustic waves. Numerical analysis of the
changes of spectra of Langmuir wave, spin-electron-acoustic wave, and
ion-acoustic wave under the change of the spin polarization of electrons is
presented. It is demonstrated that spectra of Langmuir wave and
spin-electron-acoustic wave getting closer to each other in the relativistic
limit. Spin dependence of the amplitude and width of the relativistic
spin-electron-acoustic soliton is demonstrated as well. Reformation of the
bright soliton of potential of the electric field into the dark soliton under
the influence of the relativistic effects is found.
