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recently it has become possible to produce mev electrons with short - pulse multiteravat laser system  @xcite .
the fast ignitor concept  @xcite relevant to the inertial confinement fusion enhances the interest in this process . in an underdense plasma , electrons and ions
tend to be expelled from the focal spot by the ponderomotive pressure of an intense laser pulse , and the formed channel  @xcite can act as a propagation guide for the laser beam . depending on the quality of the laser beam , the cumulative effects of ponderomotive and relativistic self focusing  @xcite can significantly increase the laser intensity . for these laser pulses ,
the laser electric and magnetic fields reach few hundreds of gv / m and megagauss , respectively , and quiver velocity in the laser field is closed to the light speed .
the component of the resulting lorentz force @xmath1 accelerates electrons in the longitudinal direction , and energies of several tens of mev can be achieved  @xcite .
recently the spectra of hot electrons ( i.e. with energy in mev region ) were investigated . in paper
@xcite the interaction of 500  fs fwhm pulses with ch target was measured .
the electrons with energy up to 20  mev were observed .
moreover for electrons with energies higher than 5  mev the change of electron temperature was observed : from 1  mev ( for energy of electrons @xmath2  5  mev ) to 3  mev ( for energy of electrons @xmath0  mev ) . in this paper
the interaction of femtosecond laser pulse with electron plasma will be investigated . within the theoretical framework of heaviside temperature wave equation , the heating process of the plasma will be described .
it will be shown that in vicinity of energy of 5  mev the sound velocity in plasma reaches the value   @xmath3 and is independent of the electron energy .
the mathematical form of the hyperbolic quantum heat transport was proposed in  @xcite and  @xcite . under the absence of heat or mass sources
the equations can be written as the heaviside equations : @xmath4 and @xmath5 for mass and thermal energy transport respectively .
the discussion of the properties of eq .
( [ eq1 ] ) was performed in  @xcite and eq .
( [ eq2 ] ) in  @xcite . in eq .
( [ eq1 ] ) @xmath6   is the velocity of density wave , @xmath7 is the diffusion coefficient for mass transfer . in eq .
( [ eq2 ] ) @xmath8 is the velocity for thermal energy propagation and @xmath9 is the thermal diffusion coefficient .    in the subsequent we will discuss the complex transport phenomena , i.e. diffusion and convection in the external field .
the current density in the case when the diffusion and convection are taken into account can be written as : @xmath10 in equation  ( 3 ) the first term describes the fourier diffusion , the second term is the maxwell - cattaneo term and the third term describes the convection with velocity  @xmath11 .
the continuity equation for the transport phenomena has the form : @xmath12 considering both equations  ( 3 ) and  ( 4 ) one obtains the transport equation : @xmath13 in equation  ( [ eq5 ] ) @xmath14 denotes the relaxation time for transport phenomena .
let us perform the smoluchowski transformation for   @xmath15 @xmath16 \rho_{1 } ( x , t).\label{eq6}\ ] ] after substituting  @xmath15 formula  ( 6 ) to equation  ( 5 ) one obtains for  @xmath17 @xmath18 considering that  @xmath19 equation  ( [ eq7 ] ) can be written as @xmath20 in the same manner equation for the temperature field can be obtained : @xmath21 in equation  ( [ eq9 ] ) @xmath22 and @xmath8 are : relaxation time for heat transfer , diffusion coefficient , heat convection velocity and thermal wave velocity .    in this paper
we will investigate the structure and solution of the equation  ( [ eq9 ] ) .
for the hyperbolic heat transport eq .
( [ eq9 ] ) we seek a solution of the form : @xmath23 after substitution of eq .
( [ eq10 ] ) into eq .
( [ eq9 ] ) one obtains : @xmath24 considering that @xmath25 eq .
( [ eq11 ] ) can be written as @xmath26 after omitting the term @xmath27 in comparison to the term @xmath28 eq .
( [ eq12 ] ) takes the form : @xmath29 considering that @xmath30 one obtains from eq .
( [ eq13 ] ) @xmath31 equation  ( [ eq14 ] ) is the master equation for heat transfer induced by ultra - short laser pulses , i.e.  when @xmath32 . in the following
we will consider the eq .
( [ eq14 ] ) in the form : @xmath33 where @xmath34 in equation  ( [ eq16 ] ) the ratio @xmath35 is the mach number for thermal processes , for @xmath36 is the sound velocity in the gas of heat carriers  @xcite .    in monograph
@xcite the structure of equation  ( 15 ) was investigated .
it was shown that for @xmath37 , i.e. @xmath38 , subsonic heat transfer is described by the modified telegrapher equation @xmath39 for @xmath40 i.e. for supersonic case heat transport is described by klein - gordon equation : @xmath41 the velocity of sound  @xmath42 depends on the temperature of the heat carriers .
the general formula for sound velocity reads  @xcite : @xmath43 in formula  ( [ eq20 ] ) @xmath44 and  _ g _ is of the form  @xcite : @xmath45 where @xmath46  is the light velocity , @xmath47  is the mass of heat carrier , @xmath48  is the temperature of the gas and @xmath49 are modified bessel functions of the second kind . in fig .
[ fig1 ] the ratio of  @xmath50 was presented are the function of  @xmath51 . fig .
[ fig1]a presents the @xmath52 for @xmath53 ( nonrelativistic approximations ) and fig .
[ fig1]b presents the @xmath54 for the very high temperature heat carriers gas , i.e. @xmath55 ( relativistic gas ) .
it is interesting to observe that for nonrelativistic gas , @xmath56 is a linear function of temperature . from formula  ( [ eq21 ] ) can be concluded  @xcite that for @xmath57 one obtains @xmath58 i.e as for maxwellian nonrelativistic gas . on the other hand for @xmath59 , @xmath60 and
is independent of  @xmath48 where @xmath61 denotes the sound velocity in the photon gas . in this paper
we will study the heat transfer in the relativistic gas , i.e. when @xmath62 is constant . in that case
( [ eq18 ] ) and ( [ eq19 ] ) are the linear hyperbolic equations . in fig .
[ fig1](c ) it is shown that maxwellian approximation is not valid for @xmath63 and moreover gives a wrong description of @xmath42 for @xmath64 for @xmath65 in complete disagreement with special relativity theory . in fig .
[ fig2](a , b , c ) the results of calculations of the sound velocity of electron gas and in fig .
[ fig3](a , b , c ) for proton gas are presented respectively . for
the initial conditions @xmath66 solutions of the equation can be find in  @xcite : @xmath67 where @xmath68 and @xmath69 the general equation for complex heat transfer : diffusion plus convection can be written as : @xmath70 considering eqs .
( [ eq6 ] ) , ( [ eq10 ] ) and ( [ eq23 ] ) the solution of equation  ( [ eq24 ] ) is @xmath71      e^{-\frac{2}{2\tau _ { t } } }      \cdot \bigg ( \frac{f(x - v_{t } t)+f(x+v_{t }      t)}{2}\\ & & \mbox { } + \frac{1}{2 } \int\limits_{x - v_{t } t}^{x+v_{t } t}\phi ( x , t , z)dz\bigg ) .
\end{aligned}\ ] ] in fig .
[ fig4 ] the comparison of the calculation of sound velocity for electron gas and the electron spectra  [ 7 ] is presented .
the change of the shape of the electron spectra in vicinity of 5  mev can be easily seen . at the total energy of the
5  mev the sound velocity in electron plasma is constant and independent of electron energy .
electrons with velocities greater than @xmath72 can generate the shock thermal waves which heats the plasma to higher temperature .     as the function @xmath73 for cold heat carriers @xmath74 ;
( b ) for hot heat carriers @xmath75 ; ( c ) comparison of the ratio @xmath76 for hot carriers ( width 1em height .8ex depth -.5ex ) , ultra - relativistic carriers ( width 1em height 1ex depth -.5ex ) and maxwellian approximation ( width 1em height .6ex depth -.5ex),height=207 ]     as the function @xmath73 for cold heat carriers @xmath74 ; ( b ) for hot heat carriers @xmath75 ; ( c ) comparison of the ratio @xmath76 for hot carriers ( width 1em height .8ex depth -.5ex ) , ultra - relativistic carriers ( width 1em height 1ex depth -.5ex ) and maxwellian approximation ( width 1em height .6ex depth -.5ex),height=207 ]     as the function @xmath73 for cold heat carriers @xmath74 ; ( b ) for hot heat carriers @xmath75 ; ( c ) comparison of the ratio @xmath76 for hot carriers ( width 1em height .8ex depth -.5ex ) , ultra - relativistic carriers ( width 1em height 1ex depth -.5ex ) and maxwellian approximation ( width 1em height .6ex depth -.5ex),height=207 ]    , but for electrons with mass m=0.51  mev / c@xmath77,height=226 ]    , but for electrons with mass m=0.51  mev / c@xmath77,height=226 ]    , but for electrons with mass m=0.51  mev / c@xmath77,height=226 ]     but for protons with mass m=0.98  gev , height=226 ]     but for protons with mass m=0.98  gev , height=226 ]     but for protons with mass m=0.98  gev , height=226 ]    w/@xmath78 .
( b ) the ratio   @xmath79 as the function of the electron temperature @xmath48  [ mev],height=188 ]    w/@xmath78 .
( b ) the ratio   @xmath79 as the function of the electron temperature @xmath48  [ mev],height=188 ]
in paper the heaviside equation for laser heating electron plasma was formulated and solved .
it was shown that for high energy electrons , with energy @xmath0  mev the sound velocity in plasma is constant and equal @xmath80 the superheating of plasma with electron energy  @xmath81  mev can be achieved by the generation of thermal shock waves .
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in this paper the superheating of electron plasma by femtosecond laser pulses is investigated . with heaviside thermal equation ( _ lasers in engineering _ , * 12 , * ( 2002 ) , p.  17 ) the generation of superhot electrons is described . 
 it is shown that in hot electron plasma ( i.e. with electron energies @xmath0  mev ) the thermal shock waves can be generated . 
 + * key words : * femtosecond laser pulses ; hot electron plasma ; shock thermal waves .    
 * sub- and supersonic heat motion + induced by + femtosecond laser pulses *    _ * _ + _ institute of electron technology _ , _ al . 
 _ _ lotnikw 32 - 46 _ , _ 02 - 628 warsaw poland . _ +   + _ institute of experimental physics and physics teacher college _ , _ warsaw university _ , _ hoa 69 _ , _ 00 - 681 warsaw _ , _ poland . _ +    to 1.5inwidth1.5 in height0.4pt depth0 in    
 _ * corresponding author _