document: diffusive acceleration at newtonian shock fronts is an extensively studied phenomenon . 
   detailed discussions of the current status of the investigations can be found in some recent excellent reviews @xcite . while much is by 
   now well understood , some issues are still subjects of much debate , for the theoretical and phenomenological implications that they may have . 
   the most important of these is the backreaction of the accelerated particles on the shock : the violation of the _ test particle approximation _ occurs when the acceleration process becomes sufficiently efficient to generate pressures of the accelerated particles which are comparable with the incoming gas kinetic pressure . 
   both the spectrum of the particles and the structure of the shock are changed by this phenomenon , which is therefore intrinsically nonlinear .    at present 
   there are three viable approaches to account for the backreaction of the particles upon the shock : one is based on the ever - improving numerical simulations @xcite that allow a self - consistent treatment of several effects . 
   the second approach is a _ fluid _ approach , and treats cosmic rays as a relativistic second fluid . 
   this class of models was proposed and discussed in @xcite . 
   these models allow one to obtain the thermodynamics of the modified shocks , but do not provide information about the spectrum of accelerated particles . 
   the third approach is analytical and may be very helpful to understand the physics of the nonlinear effects in a way that sometimes is difficult to achieve through simulations , due to their intrinsic complexity .    in ref . 
   @xcite a perturbative approach was adopted , in which the pressure of accelerated particles was treated as a small perturbation . by construction 
   this method provides an answer only for weakly modified shocks . 
   an alternative approach was proposed in @xcite , based on the assumption that the diffusion of the particles is sufficiently energy dependent that different parts of the fluid are affected by particles with different average energies . 
   the way the calculations are carried out implies a sort of separate solution of the transport equation for subrelativistic and relativistic particles , so that the two spectra must be somehow connected at @xmath0 _ a posteriori_.    recently , in @xcite , the effects of the non - linear backreaction of accelerated particles on the maximum achievable energy were investigated , together with the effects of geometry . 
   the maximum energy of the particles accelerated in supernova remnants in the presence of large acceleration efficiencies was also studied in @xcite . 
   the need for a _ 
   practical _ solution of the acceleration problem in the non - linear regime was recognized in @xcite , where a simple analytical broken - power - law approximation of the non - linear spectra was presented . 
   recently , some promising analytical solutions of the problem of non - linear shock acceleration have appeared in the literature @xcite . 
   these solutions seem to avoid many of the limitations of previous approaches . 
   numerical simulations have been instrumental to identify the dramatic effects of the particles backreaction : they showed that even when the fraction of particles injected from the thermal gas is relatively small , the energy channelled into these few particles can be an appreciable part of the kinetic energy of the unshocked fluid , making the test particle approach unsuitable . 
   the most visible effects of the backreaction of the accelerated particles on the shock appear in the spectrum of the accelerated particles , which shows a peculiar flattening at the highest energies . 
   the analytical approaches reproduce well the basic features arising from nonlinear effects in shock acceleration . 
   while several calculations exist of the nonlinear effects in the shock acceleration of quasi - monochromatic particles injected at a shock surface , there is no description at present of how these effects appear , if they do , when the shock propagates in a medium where ( pre)accelerated particles already exist . 
   the linear theory of this phenomenon was developed by bell @xcite , but has never been generalized to its nonlinear extension . 
   in fact , the backreaction can severely affect the process of re - energization of preaccelerated particles : bell already showed that for strong shocks the energy content of a region where cosmic rays were present could be easily enhanced by a factor @xmath1 at each shock passage , so that equipartition could be readily reached . in these conditions the backreaction of the accelerated particles should be expected . 
   we report here on the first analytical treatment of the shock acceleration in the presence of seed nonthermal particles , with the inclusion of their nonlinear backreaction on the shock . 
   our approach is a generalization of the analytical method introduced in @xcite to describe the nonlinear shock acceleration with monochromatic injection of quasi - thermal particles . 
   in fact , we present here also a general calculation that accounts for both thermal particles and seed nonthermal particles . 
   this case may be of interest for the study of supernova shocks propagating through the interstellar medium ( ism ) where pressure balance exists between gas and cosmic rays . 
   nonlinear effects in shock acceleration of thermal particles result in the appearance of multiple solutions in certain regions of the parameter space . 
   this behaviour resembles that of critical systems , with a bifurcation occurring when some threshold is reached . in the case of shock acceleration 
   , it is not easy to find a way of discriminating among the multiple solutions when they appear . 
   neverthless , in @xcite , a two fluid approach has been used to demonstrate that when three solutions appear , the one with intermediate efficiency for particle acceleration is unstable to corrugations in the shock structure and emission of acustic waves . 
   plausibility arguments may be put forward to justify that the system made of the shock plus the accelerated particles may sit at the critical point , but the author is not aware of any real proof that this is what happens . 
   the physical parameters that play a role in this approach to criticality are the maximum momentum achievable by the particles in the acceleration process , the mach number of the shock , and the injection efficiency , namely the fraction of thermal particles crossing the shock that are accelerated to nonthermal energies . 
   the last of them , the injection efficiency , hides a crucial physics problem by itself , and may play an important role in establishing the level of shock modification . 
   this efficiency parameter in reality is defined by the microphysics of the shock and should not be a free parameter of the problem . 
   unfortunately , our poor knowledge of such microphysics , in particular for collisionless shocks , does not allow us to establish a clear and universal connection between the injection efficiency and the macroscopic shock properties . 
   the paper is structured as follows : in  [ sec : nonlin ] we describe the effect of non linearity on shock acceleration and our mathematical approach to describe it . in particular 
   we generalize previous calculations to the case in which seed particles exist in the region where the shock is propagating ; in  [ sec : gasdyn ] we describe the gas dynamics in the presence of a non - negligible pressure of accelerated particles . in  [ sec : results ] we describe our results , with particular attention for the onset of the particle backreaction and for the appearance of multiple solutions . 
   we conclude in  [ sec : conclusions ] . 
   in this section we solve the diffusion - convection equation for the cosmic rays in the most general case in which particles are injected according to some function @xmath2 and the shock propagates within a region where some cosmic ray distribution exists , that we denote as @xmath3 , spatially homogeneous before the shock crossing ( upstream ) .    for simplicity 
   we limit ourselves to the case of one - dimensional shocks , but the introduction of different geometrical effects is relatively simple , and in fact many of our conclusions should not be affected by geometry . 
   the equation that describes the diffusive transport of particles in one dimension is @xmath4 -   u   \frac{\partial f ( x , p)}{\partial x } + \frac{1}{3 }    \frac{d u}{d x}~p~\frac{\partial f(x , p)}{\partial p } +   q(x , p ) = 0 , \label{eq : trans}\ ] ] where we assumed stationarity ( @xmath5 ) . the @xmath6 axis is oriented from upstream to downstream , as in fig . 
   the presence of pre - existing cosmic rays is introduced as a boundary condition at upstream infinity , by requiring that @xmath7 . 
   one should keep in mind that the common picture of a fluid whose speed is constant ( @xmath8 ) until it hits the shock surface is not appropriate for modified shocks . 
   in fact , in this case , the pressure of the accelerated particles may become large enough to slow down the fluid before it crosses the shock surface . 
   therefore in general at upstream infinity the gas flows at speed @xmath9 , different from @xmath8 ( fluid speed immediately upstream of the shock ) . 
   the two quantities are approximately equal only when the accelerated particles do not have dynamical relevance . 
   the injection term is taken in the form @xmath10 .    as a first step , 
   we integrate eq . 
   [ eq : trans ] around @xmath11 , from @xmath12 to @xmath13 , denoted in fig . 1 as points `` 1 '' and `` 2 '' respectively , so that the following equation can be written : @xmath14_2 - \left [ d \frac{\partial f}{\partial 
   x}\right]_1 + \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p)= 0,\ ] ] where @xmath8 ( @xmath15 ) is the fluid speed immediately upstream ( downstream ) of the shock and @xmath16 is the particle distribution function at the shock location . by requiring that the distribution function downstream is independent of the spatial coordinate ( homogeneity ) , we obtain @xmath17_2=0 $ ] , so that the boundary condition at the shock can be rewritten as @xmath14_1 = \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p ) . 
   \label{eq : boundaryshock}\ ] ] we can now perform the integration of eq . 
   ( [ eq : trans ] ) from @xmath18 to @xmath12 ( point `` 1 '' ) , in order to take into account the boundary condition at upstream infinity . 
   ( [ eq : boundaryshock ] ) we obtain @xmath19 we can now introduce a quantity @xmath20 defined as @xmath21 whose physical meaning is instrumental to understand the nonlinear backreaction of particles . 
   the function @xmath20 is the average fluid velocity experienced by particles with momentum @xmath22 while diffusing upstream away from the shock surface . 
   in other words , the effect of the average is that , instead of a constant speed @xmath8 upstream , a particle with momentum @xmath22 experiences a spatially variable speed , due to the pressure of the accelerated particles that is slowing down the fluid . since the diffusion coefficient is in general @xmath22-dependent , particles with different energies _ feel _ a different compression coefficient , higher at higher energies if , as expected , the diffusion coefficient is an increasing function of momentum . 
   the role of @xmath20 can also be explained as follows : the distribution function @xmath23 at a distance @xmath6 from the shock surface can be written as @xcite @xmath24,\ ] ] where @xmath25 is the local slope of @xmath26 and the diffusion coefficient @xmath27 has been assumed independent of the location @xmath6 . in first approximation , we can assume that the exponential factor remains important when it is of order unity , namely when its argument is much less than unity . 
   we can therefore introduce a distance @xmath28 , which is the distance at which the exponential equals one . 
   this means that @xmath29\approx u(x_p),\ ] ] so that the speed @xmath20 can be interpreted as the fluid speed at the point @xmath28 where the particles with momentum @xmath22 reverse their motion in the upstream fluid and return to the shock . 
   note that the diffusion coefficient enters the calculation of the distance @xmath28 but does not enter directly the calculation of @xmath20 . 
   in other words , different diffusion coefficients may move the point where the fluid speed is @xmath20 closer to or farther from the shock surface , but do not affect the value of @xmath20 . 
   this approach is similar to that introduced in @xcite .    with the introduction of @xmath20 , eq . 
   ( [ eq : step ] ) becomes @xmath30 + u_0 f_\infty + q_0(p ) = 0 , \label{eq : step1}\ ] ] where we used the fact that @xmath31.\ ] ] eq . 
   ( [ eq : step1 ] ) can be written in a way that resembles more the equation for shocks with no particles backreaction but in the presence of seed particles with distribution @xmath3 and injection function @xmath32 : @xmath33 - u_0 f_\infty -q_0(p ) \right\}. \label{eq : transport}\ ] ] the solution of this equation can be written in the following implicit form : @xmath34 @xmath35}{u_{\bar p } - u_2 }   \exp\left\{-\int_{\bar p}^p \frac{dp'}{p ' }   \frac{3}{u_{p ' } - u_2}\left[u_{p'}+\frac{1}{3}p ' \frac{du_{p'}}{d p'}\right ] \right\}. \label{eq : solut}\ ] ] in the case of monochromatic injection with momentum @xmath36 at the shock surface , we can write @xmath37 where @xmath38 is the gas density immediately upstream ( @xmath12 ) and @xmath39 parametrizes the fraction of the particles crossing the shock which are going to take part in the acceleration process . the injection term in eq . 
   ( [ eq : solut ] ) becomes @xmath40\right\}=\ ] ] @xmath41\right\}. \label{eq : inje}\ ] ] here we introduced the two quantities @xmath42 and @xmath43 , which are respectively the compression factor at the gas subshock and the total compression factor between upstream infinity and downstream . 
   the two compression factors would be equal in the test particle approximation . 
   for a modified shock , @xmath44 can attain values much larger than @xmath45 and more in general , much larger than @xmath46 , which is the maximum value achievable for an ordinary strong non - relativistic shock . 
   the increase of the total compression factor compared with the prediction for an ordinary shock is responsible for the peculiar flattening of the spectra of accelerated particles that represents a feature of nonlinear effects in shock acceleration . in terms of @xmath45 and @xmath44 the density immediately upstream is @xmath47 .    in eq . 
   ( [ eq : inje ] ) we can introduce a dimensionless quantity @xmath48 so that @xmath49    introducing the same formalism also for the reacceleration term in eq . 
   ( [ eq : solut ] ) , we obtain the general expression @xmath50 @xmath51    the solution of the problem is known if the velocity field @xmath48 is known . 
   the nonlinearity of the problem reflects in the fact that @xmath52 is in turn a function of @xmath16 as it is clear from the definition of @xmath20 . in order to solve the problem we need to write the equations for the thermodynamics of the system including the gas , the reaccelerated cosmic rays , the cosmic rays accelerated from the thermal pool and the shock itself . 
   we write and solve these equations in the next section . 
   the velocity , density and thermodynamic properties of the fluid can be determined by the mass and momentum conservation equations , with the inclusion of the pressure of the accelerated particles and of the preexisting cosmic rays . 
   we write these equations between a point far upstream ( @xmath18 ) , where the fluid velocity is @xmath9 and the density is @xmath53 , and the point where the fluid upstream velocity is @xmath20 ( density @xmath54 ) . 
   the index @xmath22 denotes quantities measured at the point where the fluid velocity is @xmath20 , namely at the point @xmath28 that can be reached only by particles with momentum @xmath55 . 
   the mass conservation implies : @xmath56 conservation of momentum reads : @xmath57 where @xmath58 and @xmath59 are the gas pressures at the points @xmath18 and @xmath60 respectively , and @xmath61 is the pressure in accelerated particles at the point @xmath28 ( we used the symbol @xmath62 to mean _ cosmic rays _ , in the sense of _ accelerated particles _ ) . the mass flow in accelerated particles has reasonably been neglected . 
   our basic assumption , similar to that used in @xcite , is that the diffusion is @xmath22-dependent and more specifically that the diffusion coefficient @xmath27 is an increasing function of @xmath22 . 
   therefore the typical distance that a particle with momentum @xmath22 moves away from the shock is approximately @xmath63 , larger for high energy particles than for lower energy particles increases with @xmath22 faster than @xmath20 does , therefore @xmath64 is a monotonically increasing function of @xmath22 . ] . as a consequence 
   , at each given point @xmath28 only particles with momentum larger than @xmath22 are able to affect appreciably the fluid . strictly speaking 
   the validity of the assumption depends on how strongly the diffusion coefficient depends on the momentum @xmath22 . 
   the cosmic ray pressure at @xmath28 is the sum of two terms : one is the pressure contributed by the adiabatic compression of the cosmic rays at upstream infinity , and the second is the pressure of the particles accelerated or reaccelerated at the shock ( @xmath65 ) and able to reach the position @xmath28 . 
   since only particles with momentum @xmath66 can reach the point @xmath60 , we can write @xmath67 @xmath68 where @xmath69 is the velocity of particles with momentum @xmath22 , @xmath70 is the maximum momentum achievable in the specific situation under investigation , and @xmath71 is the adiabatic index for the accelerated particles .    let us consider separately the case of a strongly modified and weakly modified shock , in order to determine the best choice for @xmath71 . in the case of strongly modified shocks , we will show that most energy is piled up in the region @xmath72 , therefore in this case we can safely adopt @xmath73 , appropriate for a relativistic gas . for weakly modified shocks , 
   the accelerated particles have an approximately power law spectrum with a slope @xmath74 . 
   it can be shown that in this case @xmath75 , so that the relativistic result @xmath73 still applies for @xmath76 ( strong shocks ) . 
   for steeper spectra ( @xmath77 ) a larger adiabatic index should be adopted , but in those cases the solution is basically independent of the choice of @xmath71 because of the weak backreaction of the particles . for the purpose of carrying out our numerical calculation we will therefore always take @xmath73 . 
   the pressure of cosmic rays at upstream infinity is simply @xmath78 where @xmath79 is some minimum momentum in the spectrum of seed particles . for simplicity , we assume that @xmath80 , namely the minimum momentum of the seed particles coincides with the momentum at which particles are injected in the shock and are accelerated .    from eq . 
   ( [ eq : pressure ] ) we can see that there is a maximum distance , corresponding to the propagation of particles with momentum @xmath70 such that at larger distances the fluid is unaffected by the accelerated particles and @xmath81 . 
   we will show later that for strongly modified shocks the integral in eq . 
   ( [ eq : cr ] ) is dominated by the region @xmath82 . 
   this improves even more the validity of our approximation @xmath83 . 
   this also suggests that different choices for the diffusion coefficient @xmath27 may affect the value of @xmath70 , but at fixed @xmath70 the spectra of the accelerated particles should not change in a significant way .    assuming an adiabatic compression of the gas in the upstream region , we can write @xmath84 where we used mass conservation , eq . 
   ( [ eq : mass ] ) . the gas pressure far upstream is @xmath85 , where @xmath86 is the ratio of specific heats for the gas ( @xmath87 for an ideal gas ) and @xmath88 is the mach number of the fluid far upstream . 
   we introduce now a parameter @xmath89 that quantifies the relative weight of the cosmic ray pressure at upstream infinity compared with the pressure of the gas at the same location , @xmath90 . 
   using this parameter and the definition of the function @xmath52 , the equation for momentum conservation becomes @xmath91 +   \frac{1}{\rho_0 u_0 ^ 2 } \frac{d{\tilde p}_{cr}}{dp } = 0.\ ] ] using the definition of @xmath92 and multiplying by @xmath22 , this equation becomes @xmath93 =   \frac{4\pi}{3 \rho_0 u_0 ^ 2 } p^4 v(p ) f_0(p ) , \label{eq : eqtosolve}\ ] ] where @xmath16 depends on @xmath52 as written in eq . 
   ( [ eq : laeffe ] ) . 
   ( [ eq : eqtosolve ] ) is therefore an integral - differential nonlinear equation for @xmath52 . 
   the solution of this equation also provides the spectrum of the accelerated particles . 
   the last missing piece is the connection between @xmath45 and @xmath44 , the two compression factors appearing in eq . 
   ( [ eq : solut ] ) . 
   the compression factor at the gas shock around @xmath11 can be written in terms of the mach number @xmath94 of the gas immediately upstream through the well known expression    @xmath95    on the other hand , if the upstream gas evolution is adiabatic , then the mach number at @xmath12 can be written in terms of the mach number of the fluid at upstream infinity @xmath88 as @xmath96 so that from the expression for @xmath45 we obtain @xmath97^{\frac{1}{\gamma_g+1}}. \label{eq : rsub_rtot}\ ] ]    now that an expression between @xmath45 and @xmath44 has been found , eq . 
   ( [ eq : eqtosolve ] ) basically is an equation for @xmath45 , with the boundary condition that @xmath98 . finding the value of @xmath45 ( and the corresponding value for @xmath44 ) such that @xmath98 also provides the whole function @xmath52 and , through eq . 
   ( [ eq : solut ] ) , the distribution function @xmath26 for the particles resulting from acceleration and reacceleration in the nonlinear regime . if the backreaction of the accelerated particles is small , the _ test particle _ 
   solution must be recovered . 
   in this section we investigate the shock modification due to the backreaction of the accelerated particles . 
   we split the discussion in two parts : in  [ subsec : seed ] we consider the case of pre - existing seed particles populating the region where the shock is propagating , and re - energized by the shock . 
   we find the conditions for the onset of the nonlinear regime in which the shock gets modified by the re - accelerated particles . 
   in particular we show that the critical behaviour found for shock acceleration and manifesting itself through the appearance of three solutions , takes place also in this case . in 
   [ subsec : thermal ] we explore the more complicated situation in which a shock accelerates a new population of particles while possibly reaccelerating a pre - existing population of seed particles .      here 
   we assume that a fluid is moving with speed @xmath9 in a region where the temperature is @xmath99 . 
   the mach number is some pre - defined value @xmath88 , which can be easily related to @xmath9 and @xmath99 . in the region of interest 
   we assume that a population of seed particles is present with energy per particle already higher than some injection energy necessary for the particles to _ feel _ the shock as a discontinuity . for simplicity 
   we assume that these seed particles have a spectrum which is a power law in momentum in the form : @xmath100 in this section we explore the situation in which the shock reacceleration of a pre - existing population of cosmic rays can modify the shock structure , but there is no acceleration of particles different from those that are already present as seed particles . 
   this is equivalent to turn off the injection term ( @xmath101 in eq . 
   ( [ eq : laeffe ] ) ) . in the next section 
   we discuss what happens when both components are present . 
   the crucial difference between the two components is that the seed particles are by definition already above the threshold for _ injection _ , so that there is no injection efficiency that instead represents such a crucial parameter for the case of acceleration of particles extracted from the thermal pool . 
   having in mind the case of shocks propagating in the interstellar medium of our galaxy , we consider here the case in which the inflowing gas and the seed particles are in pressure equilibrium , namely @xmath102 . in terms of the parameter @xmath103 
   this implies @xmath104 . 
   the results can then be easily repeated for a generic value of @xmath89 . 
   ( [ eq : eqtosolve ] ) gives the quantity @xmath52 as a function of the momentum @xmath22 for any choice of @xmath45 and @xmath44 , while the acceptable solutions are those with the right matching conditions at @xmath70 . 
   for simplicity , let us assume that the @xmath70 in eq . 
   ( [ eq : f_infty ] ) is the same maximum momentum that particles injected at the shock would achieve : this value only depends on the environmental conditions ( energy losses of the particles ) and/or on the geometry of the shock , which may allow the escape of the particles . 
   the momentum @xmath70 is the same as in the distribution of the pre - exisiting seed particles if , for instance , the seed particles have been accelerated by a shock identical to the one we are considering . in any case , this assumption is not needed for the validity of our conclusions and may be easily relaxed , it simply serves to avoid parameter proliferation . within this assumption , 
   particles re - energized at the shock are simply redistributed in the momentum range between @xmath36 and @xmath70 . 
   the solution , namely the right pair of values for the compression parameters @xmath45 and @xmath44 [ related through eq . 
   ( [ eq : rsub_rtot ] ) ] , is obtained when the solution corresponding to @xmath98 is selected .    in fig . 
   [ fig : m150 ] , we plot @xmath105 as a function of @xmath44 for a shock having mach number at infinity @xmath106 
   . the different lines are labelled by a number representing the @xmath107 of @xmath70 in units of @xmath108 . in other words 
   the parameter @xmath70 changes between @xmath109 and @xmath110 . 
   the physical solutions are found by determining the intersections of each curve with the horizontal line corresponding to @xmath98 . 
   the minimum momentum of the seed particles is taken as @xmath111 . 
   the intersection at @xmath112 is close to the well known linear solution , namely the solution that one would obtain in the test particle approximation . 
   the energy channelled into the nonthermal particles for this solution is very small , and the shock remains approximately unmodified . increasing the value of @xmath70 , the solution moves toward slightly larger values of @xmath44 ( namely the shock becomes more modified ) . 
   for some values of @xmath70 , multiple solutions appear . 
   in particular three regions of @xmath70 can be identified :    \1 ) @xmath113 : in this region ( low values of @xmath70 ) the only solution is very close to the one obtained in the test particle approximation . for the values of the parameters in fig . 
   [ fig : m150 ] , @xmath114 . 
   \2 ) @xmath115 : in this region ( intermediate values of @xmath70 three solutions appear , two of which imply a strong modification of the shock , namely an appreciable part of the energy flowing through the shock is converted into energy of the accelerated particles . for the values of the parameters in fig . [ 
   fig : m150 ] , @xmath116 . 
   \3 ) @xmath117 : in this region ( high values of @xmath70 ) the solution becomes one again , and the shock is always strongly modified ( @xmath118 ) . 
   this critical behaviour appears also when one fixes the maximum momentum and uses the mach number of the shock as the order parameter . 
   the results are shown in fig . 
   [ fig : pmax10_5 ] , where mach numbers between 10 and 500 have been considered , at fixed @xmath119 . 
   one can see that for mach numbers below @xmath120 there is only one solution . 
   for mach numbers between @xmath120 and @xmath121 three solutions appear , one of which roughly corresponds to an unmodified shock . for larger mach numbers , only this linear solution remains , and the shock is always only weakly modified .    the same situation is plotted in fig . 
   [ fig : pmax10_5rsub ] , where instead of @xmath44 on the x - axis there is @xmath122 , and @xmath45 is the compression coefficient at the gas subshock . for unmodified shocks 
   one expects @xmath123 .    again 
   three regions can be identified , in the parameter @xmath88 . 
   \1 ) for @xmath124 only one solution exists and does not necessarily correspond to unmodified shocks . 
   in fact one can see that for @xmath125 , the compression coefficient at the subshock is @xmath126 , and @xmath127 , for the situation plotted in figs . 
   [ fig : pmax10_5 ] and [ fig : pmax10_5rsub ] . 
   even smaller values of @xmath88 do not give a perfectly unmodified shock . 
   \2 ) for @xmath128 three solutions appear , two of which imply a strong modification of the shock . for the parameters used in figs . 
   [ fig : pmax10_5 ] and [ fig : pmax10_5rsub ] , @xmath129 . 
   \3 ) for @xmath130 only the nearly unmodified solution exists .    in order to emphasize the fact that the multiple solutions actually correspond to physical solutions with very different spectral characteristics for the accelerated particles we plot in fig . 
   [ fig : spectra ] the spectra for @xmath131 ( for which there is only one solution ) and @xmath106 ( for which there are three solutions ) . in both cases we used @xmath132 .    the solution corresponding to mach number @xmath131 has @xmath133 and @xmath134 , resulting in a total pressure of the accelerated particles @xmath135 ( solid line in fig . 
   [ fig : spectra ] ) . 
   the three dashed lines in fig . [ fig : spectra ] are for @xmath106 and represent the spectra for the three solutions . in numbers , these solutions ( from top to bottom in fig . 
   [ fig : spectra ] ) are summarized as follows :    _ first solution : _ @xmath136 , @xmath137 , @xmath138    _ second solution : _ 
   @xmath139 , @xmath140 , @xmath141    _ third solution : _ 
   @xmath142 , @xmath143 , @xmath144    in the cases of strong shock modification , the asymptotic spectra for @xmath145 are @xmath146 , while the linear theory , in case of strong shocks ( @xmath147 ) , would predict @xmath148 . in the region of very low energies , 
   the spectra of the reaccelerated particles tend to zero , as known from the linear theory as well . 
   nonlinear effects in shock acceleration were first investigated in the case when a fraction of the thermal gas crossing the shock surface is energized to nonthermal energies . in this section 
   we wish to apply the analytical method discussed in 
   [ sec : nonlin ] and  [ sec : gasdyn ] to the most general case in which thermal particles are accelerated but seed particles may already be present in the environment .    we start our discussion with the case of acceleration of particles from the thermal distribution , in order to show that the multiple solutions already found in other analytical approaches @xcite are also obtained by adopting the approach illustrated here . in fig . 
   [ fig : noseed ] , we plot the @xmath105 as a function of @xmath44 for the case in which a fraction @xmath39 of the particles ( as indicated in the plots ) crossing the shock is actually accelerated to suprathermal energies . 
   the mach number is chosen to be @xmath106 and the maximum momentum is taken as @xmath149 . 
   one can easily see that @xmath150 , the value for unmodified shocks , for small values of @xmath39 ( low efficiencies ) , while @xmath44 increases above 4 for @xmath151 . 
   three solutions appear for intermediate efficiencies , while the solution for the accelerated particles always predicts a strongly modified shock for high efficiencies , @xmath152 . 
   an asymptotic value of @xmath153 is achieved for the parameters used here . 
   this example demonstrates that the critical behaviour shown to appear for high values of @xmath39 in several analytical or semi - analytical calculations is in fact also predicted by the approach proposed here . 
   next question to ask is however whether the presence of seed particles can change the critical behavior of shocks . in order to answer this question 
   we consider a situation in which particles are accelerated from the thermal pool with efficiency @xmath39 , and at the same time the shock propagates in a medium where the preshock pressure in seed particles equals the gas pressure . 
   this situation is considered to resemble that of a supernova explosion in our galaxy , where cosmic rays fill the volume remaining in quasi - equipartition with the gas . in order to test the critical structure of the shock 
   we calculate @xmath105 as a function of the compression factor @xmath44 between upstream infinity and downstream . 
   our results are plotted in fig . 
   [ fig : crseed ] . 
   the lines refer to the cases @xmath154 from bottom to top , as in fig . 
   [ fig : noseed ] . 
   it is clear from this plot that the shock may be modified by the backreaction of the accelerated particles even for very low values of @xmath39 , because of the presence of seed particles . in other words , 
   the nonlinearity of the shock can well be dominated by the presence of preaccelerated particles rather than by the acceleration of particles from the thermal pool . 
   clearly this depends however on the values of the parameters ( injection momentum , @xmath39 , maximum momentum achievable , mach number ) . 
   the injection momentum , in principle , could be related to @xmath39 in order to reduce the number of free parameters of the problem . 
   this is possible in the assumption that the particles downstream keep a thermal distribution . 
   simulations suggest that the particles injected in the accelerator are the ones with momentum a few times the thermal momentum of the particles downstream , so that @xmath39 can be simply calculated by integration of the maxwell - boltzmann ( mb ) distribution above this minimum momentum . 
   unfortunately , we do not know whether the particle distribution downstream is in fact maxwell - boltzmann - like . 
   moreover , we do not know exactly what is the threshold to impose on the injection momentum , which is a delicate issue because the number of particles taking part to the acceleration process would result from the integral of the mb distribution over its exponentially decreasing part 
   . therefore we preferred here to keep @xmath39 and the injection momentum as separate free parameters . 
   we proposed a semi - analytical approach to show that the backreaction of particles accelerated at a shock is able to affect the shock itself , in such a way that the shock and the accelerated particles become parts of a nonlinear system . 
   in particular , for the first time we included in this kind of calculations the seed particles that may be present in the region where the shock is propagating and that can be re - energized by the shock .    while a test - particle approach to this problem was first presented in @xcite , a nonlinear treatment was never investigated . in the pioneering work of ref . 
   @xcite , it was recognized that the energy of the seed particles could be enhanced by about one order of magnitude at each shock passage , and that after an infinite number of strong shocks passing through the region , the spectrum of the particles would tend to the asymptotic spectrum @xmath155 . 
   two comments are in order . 
   first , the continuous increase of the cosmic ray energy due to the re - energization of seed particles leads unavoidably the shock to be modified by the nonthermal pressure , unless one starts from an uninterestingly small pressure of seed particles at the beginning . 
   second , the fact that the spectrum becomes flatter than @xmath156 , which is the result for a strong non - relativistic shock , implies that most of the nonthermal energy is pushed to the highest energies , therefore the shock again can be more easily modified . 
   both these points suggest that a nonlinear treatment of shock re - acceleration is required .    in our galaxy , cosmic rays 
   are observed to be in rough equipartition with the gas pressure and with magnetic fields , therefore supernova shocks or shocks generated in other environments propagate in a medium in which the seed particles ( cosmic rays ) are non - negligible . in these circumstances 
   the non - linear effects may be very important . 
   we showed here that in fact for some regions of the parameter space , the shock is modified mainly by the backreaction of the seed particles rather than by the cosmic rays accelerated at the shock from the thermal pool . 
   the spectra of the re - accelerated particles have also been calculated . 
   the interesting phenomenon of the appearance of multiple shock solutions , already known for the case of shock acceleration , appears also for the case of reacceleration . 
   this puzzling phenomenon may suggest that the shock behaves as a self - regulating system settling on the critical point , as proposed in @xcite . on the other hand , 
   it is possible that the multiple solutions may be the artifact of some of the assumptions used in the analytical approach , in particular the request for time independent ( stationary ) solutions and the fact that the role of the self - generated waves on the diffusion coefficient is not taken into account . 
   further investigation , in particular in the direction of a detailed comparison of our results with numerical simulations of shock acceleration is required in order to unveil the physical meaning of the multiple solutions for modified shocks .    from the phenomenological point of view , it would be certainly worth to study the implications of nonlinear shock reacceleration on the nonthermal activity in astrophysical environments where the effect is expected to play an important role , in particular in the case of supernova remnants . in particular , as pointed out by the referee , the suggested dominance of reaccelerated ambient seed particles over freshly injected particles may have serious implications for the spectra of secondary nuclei resulting from spallation processes . 
   the author gratefully acknowledges useful discussions with m. baring , d. ellison and m. vietri . 
   the author is also grateful to the referee , l.oc . 
   drury for his very useful remarks . 
   this work was partially supported through grant cofin 2002 at the arcetri astrophysical observatory .
summary: particles crossing repeatedly the surface of a shock wave can be energized by first order fermi acceleration . 
   the linear theory is successful in describing the acceleration process as long as the pressure of the accelerated particles remains negligible compared to the kinetic pressure of the incoming gas ( the so - called test particle approximation ) . when this condition is no longer fulfilled , 
   the shock is modified by the pressure of the accelerated particles in a nonlinear way , namely the spectrum of accelerated particles and the shock structure determine each other . 
   in this paper we present the first description of the nonlinear regime of shock acceleration when the shock propagates in a medium where seed particles are already present . 
   this case may apply for instance to supernova shocks propagating into the interstellar medium , where cosmic rays are in equipartition with the gas pressure . 
   we find that the appearance of multiple solutions , previously found in alternative descriptions of the nonlinear regime , occurs also for the case of reacceleration of seed particles . 
   moreover , for parameters of concern for supernova shocks , the shock is likely to turn nonlinear mainly due to the presence of the pre - existing cosmic rays , rather than due to the acceleration of new particles from the thermal pool . 
   we investigate here the onset of the nonlinear regime for the three following cases : 1 ) seed particles in equipartition with the gas pressure ; 2 ) particles accelerated from the thermal pool ; 3 ) combination of 1 ) and 2 ) . 
   cosmic rays , high energy , acceleration