Discussion of the Results
We have derived hydrodynamic equations describing the transport of massless electrons in graphene. A linear energy spectrum of carriers should be taken into account from the very beginning of derivation. It cannot be introduced as a small correction to the parabolic dispersion (like p4 terms in Si, Ge, A3B5 ). On the other hand, the hydrodynamic equations for ultrarelativistic plasmas cannot be also directly applied to graphene as the Fermi velocity is much smaller compared to that of light.
The dependence of particle density n on drift velocity и [(Eq. (9.3)] may look confusing. However, it is the immediate consequence of the particular choice of distribution function (9.1). At the same time, one can choose the distribution function in the form
This function turns collision integral to zero, reduces to the equilibrium Fermi function at и —> 0, and the corresponding particle density does not depend on и (but the internal energy still does). It is easy to show that Euler and continuity equations derived with this function and written in terms of n and и coincide with Eqs. (9.11) and (9.10). The equation of state also holds its view. The boundary conditions for hydrodynamic equations are imposed on measurable quantities n and u. Hence, the solutions of hydrodynamic equations do not depend on the choice between distribution functions (9.1) and (9.44).
In the obtained hydrodynamic equations for electrons in graphene, the effect of linear spectrum is clearly visible. First, the drift velocity и cannot overcome the Fermi velocity Up. Secondly, a
varying fictitious hydrodynamic mass M ~(/J / v2)/-yjl-u2 / vj originates in the Euler equation. The obtained equations are neither Lorentz nor Galilean invaraint, which is directly revealed in the spectra of plasma waves in the presence of steady electron flow [Eq. (9.36)]. Our main conclusions concerning the spectra can be verified experimentally using the techniques of plasmon nano-imaging  in gated graphene under applied bias.
As we became aware recently, the spectra of plasma waves in the presence of steady flow and Dyakonov-Shur instability in graphene were analyzed in Ref. . The form of Euler equation used was different from our Eq. (9.16) even in the limit и « uF; particularly, in Ref.  the gradient term udxu did not vanish for degenerate electron systems. This led to different expression for plasma-wave velocities (3u0/4 ± u0 instead of our s±), and to higher estimate of the plasma- wave instability increment. One possible reason for the distinction of the Euler equations lies in dissimilar expressions for the electron plasma pressure P, which is substantially velocity dependent [see Eqs. (9.A2) and (9.A3): P = e/2 ~ p?/(l - P2)5'2].
The set of problems which could be solved via nonlinear hydrodynamic equations is not restricted within plasma waves. It would be also interesting to study the effects of velocity saturation associated with the upper limit of drift velocity и equal to uF. Those effects could be pronounced in graphene samples on substrates with high optical phonon energy. If it is the case, the velocity saturation caused by emission of optical phonons  seems as irrelevant.
For rigorous simulation of emerging graphene-based devices for THz generation and detection  one can as well employ the derived nonlinear equations. In the case, however, an Euler equation for holes and electron-hole friction terms should be supplied . With large electron mobility and new hydrodynamic nonlinearities, graphene-based THz devices could outperform those based on conventional semiconductors.
The hydrodynamic equations governing the collective motion of massless electrons in graphene were derived. The validity of those equationsisnotrestricted to small driftvelocities. A variable fictitious mass depending on density and velocity arises in the hydrodynamic equations. It results in several nonlinear terms specific to graphene.
The possibility of soliton formation in electron plasma of the gated graphene was shown. The quasirelativistic terms in the dynamic equations set an upper limit of the soliton amplitude and stabilize its shape.
The obtained hydrodynamic equations demonstrate a lack of Galilean and true Lorentz invariance. This noninvariance is pronouncedly revealed in the spectra of plasma waves in the presence of steady flow with velocity u0. The difference in velocities of forward and backward waves turns out to be u0 instead of 2u0, expected for massive electrons in conventional semiconductors.
The possibility of plasma-wave self-excitation in high-mobility graphene samples under certain boundary conditions (Dyakonov- Shur instability) was demonstrated. The increment of such instability in graphene is less than that in common semiconductors due to smaller difference in velocities of foiward and backward waves. However, the high mobility of electrons in graphene allows plasma-wave self-excitation for micron-length and shorter channels.