Nonlinear density waves in the single-wave model

PHYSICS OF PLASMAS 18, 032305 2011 Nonlinear density waves in the single-wave model Kiril B. Marinov1,2 and Stephan I. Tzenov2,3,a 1 2 ASTeC, STFC Daresbury Laboratory, Keckwick Lane, Daresbury, Cheshire WA4 4AD, United Kingdom The Cockcroft Institute, Keckwick Lane, Daresbury, Cheshire WA4 4AD, United Kingdom 3 Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Received 18 November 2010; accepted 16 February 2011; published online 22 March 2011 The single-wave model equations are transformed to an exact hydrodynamic closure by using a class of solutions to the Vlasov equation corresponding to the waterbag model. The warm fluid dynamic equations are then manipulated by means of the renormalization group method. As a result, amplitude equations for the slowly varying wave amplitudes are derived. Since the characteristic equation for waves has in general three roots, two cases are examined. If all the three roots of the characteristic equation are real, the amplitude equations for the eigenmodes represent a system of three coupled nonlinear equations. In the case where the dispersion equation possesses one real and two complex conjugate roots, the amplitude equations take the form of two coupled equations with complex coefficients. The analytical results are then compared to the exact system dynamics obtained by solving the hydrodynamic equations numerically. © 2011 American Institute of Physics. doi:10.1063/1.3562878 I. INTRODUCTION The processes of pattern and coherent structure formation in plasmas and plasmalike media have attracted attention for many years. A number of approximations and simplified models have been proposed, among which the singlewave model is one of the most efficient approaches to study the weakly nonlinear behavior in marginally stable plasmas. Starting from the general Vlasov–Maxwell equations for the phase space density distribution and the self-consistent electromagnetic field, one usually derives a coupled set of equations describing the evolution of the coarse-grained distribution function and a certain isolated marginally stable wave mode of the electrostatic potential. Using the method of matched asymptotic expansions, the single-wave model equations have been recently derived1 in the most general case independent of the equilibrium. A rigorous derivation of the nonlinear evolution of an unstable electrostatic wave for a multispecies Vlasov plasma has been given by Crawford and Jayaraman2 and this general physical picture has been shown to correspond to the single-wave model. Among earlier pioneering work utilizing the singlewave model, the studies dedicated to the beam-plasma instability3–5 and the bump-on-tail instability6–8 should be mentioned. The influence of the finiteness of the number of particles coupled to a monochromatic wave in a collisionless plasma has been investigated by Firpo et al.9,10 Based on the utilization of the single-wave picture some analogies between two classical models of the single-pass free electron laser dynamics and of the beam-wave plasma instability have been discussed by Antoniazzi et al.11 This implies that the results presented in the present paper will be useful to describe processes of formation of patterns and coherent structures in single-pass free electron lasers. The purpose of the present paper is not to argue on the a validity or the justification of the single-wave picture rather than to use it as a starting point for deriving amplitude equations for the slowly varying wave envelopes SVWEs . It is organized as follows. In Sec. II, we cast the selfconsistent single-wave model equations derived by del-Castillo-Negrete1 in an equivalent form by using a class of exact solutions to the Vlasov equation corresponding to the waterbag model.12,13 Further, the exact closure of hydrodynamic equations is manipulated following the renormalization group RG approach.13 As a result, amplitude equations for the SVWEs are derived. Depending on the character of the solutions of the dispersion equation, two cases can be distinguished. In the first case, where all three roots of the dispersion equation are real, the amplitude equations represent a system of three coupled nonlinear equations, describing the process of three-wave interaction and mixing. If the dispersion equation possesses one real and two complex conjugate roots, the amplitude equations take the form of two globally coupled nonlinear equations with complex coefficients. In Sec. V, the predictions of the SVWE equations are benchmarked to the exact numerical solution of the hydrodynamic equations. Finally, in Sec. VI we draw some conclusions and outlook. II. THEORETICAL MODEL AND BASIC EQUATIONS The basic equations derived by del-Castillo-Negrete,1 which will be the starting point of our subsequent analysis, can be written as tf + V xf + x Vf = 0, 1 2 3 = a t eix + a t e−ix , da + ila = i fe−ix , dt where the operator averaging is denoted by Electronic mail: s.tzenov@lancs.ac.uk. 1070-664X/2011/18 3 /032305/7/$30.00 18, 032305-1 © 2011 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp 032305-2 K. B. Marinov and S. I. Tzenov 2 Phys. Plasmas 18, 032305 2011 ¯ = 1 2 dV − 0 dx ¯ . 4 da + ila = i dt e−ix , 13 The independent time t and spatial x variables, as well as the dependent ones f x , V ; t and x ; t entering the above equations, have been properly nondimensionalized.1 For example, the dimensionless time t scales as eT, where e is the electron plasma frequency and T is the real time. Moreover, the velocity V in the equations above scales as V / VD, where VD is a velocity characteristic for the particular problem and can be chosen arbitrarily. For present purposes, we restrict the analysis to a special class of exact solutions to Eq. 1 corresponding to the waterbag distribution,12,13 f x,V;t = C H V − v− x;t − H V − v+ x;t , 5 where now the operator averaging specified by Eq. 4 involves integration on the normalized spatial variable x only, and the potential is given by expression 2 as before. We further scale the hydrodynamic and field variables according to the expressions = 0 + R, v = v0 + V, a= , = , 14 where is a formal small parameter and will be set equal to 1 at the end of the calculations. Furthermore, 0 = const and v0 = const represent the stationary solutions of Eqs. 10 and 11 , provided the stationary field amplitude a0 = 0. The basic macroscopic fluid and electrostatic field equations can be rewritten as tR where H denotes the well-known Heaviside function, C is a normalization constant, and 0 x 2 is a normalized spatial variable. It simply means that the phase space density f x , V ; t remains constant within a region confined by the boundary curves v x ; t , which are assumed to be single valued. The latter distort nonlinearly during the evolution of the system as specified by Eqs. 1 – 3 . It is convenient to introduce the macroscopic fluid variables, = − + 0 xV + v 0 xR = − 2 0v T xR x RV , =− 15 V2 2 + vTR2 , 16 2 17 18 tV + v 0 xV + 2 − x x d + il = i Re−ix , dt = t eix + t e−ix . dVf x,V;t = C v+ − v− , C 2 6 To simplify the above system of equations, we perform a Galilean transformation specified by z = x − v0t, t = t e iv0t 19 v= − dVVf x,V;t = 2 v+ − 2 v− , 7 and cast our basic system of equations in the form tR where x ; t and v x ; t are the density and the current velocity, respectively. It can be shown that the higher moments defining the particle pressure P x ; t and the heat flow Q x ; t can be expressed as P= − + 0 zV =− z RV , =− V2 2 + v TR 2 , 2 20 21 tV + 2 2 0v T zR − z z dV V − v 2 f x,V;t = C 12 v+ − v− 3 , 8 d + iL dt = = i Re−iz , t e−iz, L = l − v0 . 22 23 Q= − dV V − v 3 f x,V;t = 0. 9 t eiz + In particular, expression 9 implies that the waterbag distribution yields an exact closure of the hydrodynamic equations in the form t tv Eliminating V from the left-hand sides of Eqs. 20 and 21 , we arrive at the basic system 2 tR − 2 2 zR − 0 =− t z RV + 2 0 z V2 2 + v TR 2 , 2 24 + x v = 0, x 2 10 = x 2 + v xv + v T , 11 t + iL = i Re−iz , 2 =2 2 2 0v T 25 where 2 vT = for the subsequent analysis using the RG approach. 1 8C2 12 III. RENORMALIZATION GROUP REDUCTION OF THE MACROSCOPIC EQUATIONS is the normalized thermal speed squared. The macroscopic fluid Eqs. 10 and 11 must be supplemented with the equation for the amplitude of field mode 3 , written in the form Following the standard procedure of the RG method,13 ˆ we represent G z ; t as a perturbation expansion, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp 032305-3 Nonlinear density waves in the single-wave model Phys. Plasmas 18, 032305 2011 ˆ G z;t = n=0 nˆ Gn z;t , 26 t 1 + iL 1 = i R1e−iz . 36 ˆ where G = R , V , represents all hydrodynamic and field variables. To zero order, the perturbation equations Eqs. 24 and 25 can be written as 2 t R0 Here and afterward, the symbol “c.c.” stands for the complementary complex conjugate counterpart, while Gmn is a symmetric matrix given by the expression Gmn = m + n 2 +2 m n + 2 . 37 − 0 2 2 z R0 = 0 0e iz + 0e −iz , 27 28 It can be verified in a straightforward manner that the general solution of the first order equations is R1 = 1 t + iL 0 = i R0e−iz . B + ei + 1 0 m,n z+ t + 1 B − ei m+ n t+2iz z− t Since the equation for R0 contains only the first harmonic with regard to the spatial variable z proportional to exp iz , its solution is sought in the form R0 z;t = F t eiz + F t e−iz , where the function F satisfies the equation 2 t FmnAmAnei + c.c., 38 29 where Fmn = m + + 2 t + iL F − i 0F n m 2 +2 n 2 m = 0. 30 + −4 n+ 2 2 39 Taking the latter into account, we can write R0 z;t = m A me i mt+iz + m Ame−i mt−iz , 31 where the sum spans over all roots of the characteristic equation 2 − 2 +L + and B are the arbitrary constant amplitudes of the solution to the homogeneous part of Eq. 35 . The summation over runs from − to excluding = 1. The first harmonic proportional to exp iz describes the coupling between the fluid dynamic and the electric fields and has been taken into account in zero order. For the first order current velocity V1 we obtain V1 = − 0 1 0 = 0. 32 B + ei 1 2 0 m,n z+ t In general, the characteristic equation has three roots. However, as it will become clear from the subsequent exposition, in a number of cases depending on the particular values of physical parameters one of the roots is real, while the other two are complex conjugate. Note also that the arbitrary to this end amplitudes Am are constants with respect to the spatial z and time t variables. Using Eqs. 21 and 28 , we find V0 = − 1 0 m mA me i mt+iz − 1 B − ei + c.c., z− t − where Vmn = VmnAmAnei m+ n t+2iz 40 m n + − 1 0 m + 2 2Fmn . m+ n 2 41 mA me −i mt−iz , 33 0 = 1 0 m 2 − 2 m A me i mt . 34 Note that in first order the electric field vanishes 1 = 0. The final step in our perturbative procedure is to obtain the secular second-order solution. Retaining terms giving rise to secular contributions in the second-order solution, we can express the constitutive equations as follows: 2 t R2 It is worthwhile mentioning that in general, the solution of Eq. 27 would contain all other harmonics starting with the second one proportional to exp inz , where n is integer . They are “sound” waves, propagating at speed , and are decoupled from the electric field amplitude . Their amplitudes are vanishing in the zero order approximation. This assertion will be verified numerically and will be discussed in more detail in Sec. V. In first order the basic equations Eqs. 24 and 25 can be expressed as 2 t R1 − 2 2 z R2 − 0 2 i n+ l− m t+iz =− 1 2 0 m,n,l nlmAmAnAle + c.c., 42 t 2 + iL 2 = i R2e−iz , mnl 43 is given by the mFnl . where the coupling coefficients expression nlm − 2 2 z R1 = − n l + 2 + 3 2Fnl + n + l − m 44 0 1 m+ n t+2iz =− 1 0 m,n GmnAmAnei + c.c., 35 Taking into account terms driving self-modulation and cross modulation, it is straightforward to verify that the secular second-order solution can be expressed as Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp 032305-4 K. B. Marinov and S. I. Tzenov Phys. Plasmas 18, 032305 2011 R2 = where mn it 2 2 m,n 0 Vgm mnAm ˜ A n 2e i mt+iz + c.c., 45 1 + 2 =2 3 + . 53 = m mn + 2i m n +L mn +L = mnn 46 Apart from the self- and cross-modulation terms in Eq. 51 , additional terms describing wave mixing will be present. Therefore, in the most general case the renormalization group equation can be written as i tA 1 = − Vg1 2 2 0 11 and m A1 2 + 2 12 A2 2 + + 13 A3 2 A1 = Im m , ˜ A m = A me − mt . 47 =− Vg1 2 2 0 2 3 + + 3 2 1 2 F33 A2A2e−i t , 3 54 Moreover, the quantity Vgm is the group velocity defined as follows. Let us introduce the dispersion function13 D k, = 2 2 k − 2 +L − 0, 48 i tA 2 = − =− Vg2 2 2 0 21 corresponding to a general solution of Eq. 24 proportional to exp i t + ikz , where k is the wave number. Obviously, the dispersion equation for k = 1, that is, D 1 , = 0, reduces to the characteristic equation Eq. 32 . The group velocity is given by the expression13 Vgm = d m dk =− k=1 A1 2 + 2 22 A2 2 + + 23 A3 2 A2 Vg2 2 2 0 2 3 + + 3 2 1 2 F33 A1A2e−i t , 3 55 D k 2 2 m D m −1 49 k=1 i tA 3 = − =− Vg3 2 2 0 31 A1 2 + + 2 32 A2 2 + 33 A3 2 A3 or in an explicit form Vgm = 2 2 m 0 Vg3 2 2 0 2 3F13 1 2 − 2 2 −3 − 2L . m 50 + + 3 2F12 56 The last step is to collect all terms corresponding to increasing orders, which contribute to R z ; t and perform a resummation such as to absorb secular terms proportional to various powers of the time variable t present in R2. Since this approach is standard,13 we omit details here. IV. THE RENORMALIZATION GROUP EQUATION A 1A 2A 3e i t . Following the procedure13 of the RG method, we finally obtain the desired RG equation, Vgm ˜ i tA m = − 2 2 0 mnAm n ˜ A 2, ˜ n 51 ˜ where now Am denotes the renormalized complex amplitude of type 47 . In terms of the renormalized wave envelopes up to first order in the formal expansion parameter, the macroscopic density z ; t can be expressed as z;t = 0 The terms proportional to nn describe the effects of phase self-modulation, whereas those proportional to mn with m n are responsible for phase cross modulation. Both result in an intensity-dependent nonlinear frequency shift of the eigenfrequencies m, an effect similar to the nonlinear wave-number shift in nonlinear optics.14 In contrast, the last terms in each of the equations above describe four-wave mixing FWM effects, responsible for energy exchange between the three modes. These effects play a role only when the phase mismatch parameter is close to zero. Next, we consider one real 1 root and two 2 and 2 complex conjugate roots. In this case the renormalization group Eq. 51 takes the form i tA 1 = − Vg1 2 2 0 11 A1 2 + 12 1 + 2i 12 ˜ A2 2 A1 , 57 + m ˜ Am t ei Re m t+iz + m ˜ Am t e−i Re m t−iz . Vg2 ˜ i tA 2 = − 2 2 0 21 A1 2 + and 22 22 1 + 2i 22 ˜ ˜ A2 2 A2 , 58 52 Similar expressions hold for the current velocity V compare with Eq. 33 and for the electrostatic potential compare with Eq. 34 . It was mentioned in Sec. III that since the characteristic equation Eq. 32 is an algebraic equation of third order, it possesses three roots in general. Thus, we can distinguish the following two cases of physical interest. All three roots— 1, 2, and 3—are real. As it will be shown in Sec. V, at certain value of the parameter v0 they satisfy the resonance condition where the quantities 12 = 1 2 12 are given by the expressions 2 +L , 22 = 2 +L . 59 An important comment is now in order. The equations for the slowly varying wave amplitudes describing the coupling between the hydrodynamic and electric fields were obtained in third order in the formal small parameter. Note that reup to this order the sound waves with amplitudes B main uncoupled with the basic modes with amplitudes Am. Such coupling similar to the wave mixing of the three Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp 032305-5 Nonlinear density waves in the single-wave model Phys. Plasmas 18, 032305 2011 σ=−1, l=−1, v0=0, a(0)=0, ε=0.001 −2 −4 −6 log|a(t)| −8 log|a(t)|=0.425t−7.038 −10 Im(ω2)=−Im(ω3)=0.4248 −12 −14 0 σ=1, l=1, v =0, a(0)=0, ε=0.001 0 0.6 0.5 |a(ω)| 0.4 0.3 0.2 0.1 2 4 6 t 8 10 12 ω1=1.618 ω =−0.618 2 ω =−2 3 0 −2.5 −2 −1.5 −1 −0.5 0 ω 0.5 1 1.5 2 2.5 FIG. 1. Color online Continuous line: time dependence of the amplitude a t at l = −1 and = −1 one real and two complex roots of the characteristic equation obtained by numerical integration of Eqs. 10 , 11 , and 13 . The asymptotic behavior of a t is given by the dashed line. Its slope is the absolute value of the imaginary part of the complex roots. FIG. 2. Color online Spectrum a of the amplitude a t in the case of l = 1 and = 1 obtained from the numerical solution of Eqs. 10 , 11 , and 13 . The three peaks correspond to the three real roots of the characteristic 3. equation. Note the absence of sound waves at frequencies = eigenmodes may take place provided a resonance condition of the form m + n = 2 is fulfilled, where m and n are any two roots of the characteristic equation Eq. 32 . Note also that the amplitudes of the sound waves remain undefined to third order in the formal expansion parameter. To obtain the corresponding amplitude equations for these modes, one has to go beyond third order, which is out of the scope of the present paper. V. NUMERICAL SIMULATIONS In this section we present a comparison between the numerical solution of the system of hydrodynamic equations Eqs. 10 and 11 supplemented with the equation for single-wave amplitude 13 and the RG solution obtained in Sec. IV. Since the nondimensionalization of the initial phase space variables allows an arbitrary velocity scale, it is convenient to define the latter such that 2 vT = 2 . 3 60 In order to assess the accuracy of the model Eqs. 54 – 56 the predictions of the latter are benchmarked to the numerical solution of the original system of Eqs. 10 , 11 , and 13 . To achieve this an initial condition of the form x ; t = 0 = 1 + cos x , with in the range of 0.001–0.01 and v x ; t = 0 = v0 = const x , has been used. Equations 31 , 33 , and 34 relate the initial distributions of x ; t = 0 , t = 0 to the three slowly varying ampliv x ; t = 0 , and in the tudes A1 t = 0 , A2 t = 0 , and A3 t = 0 . Varying range of 10−3 – 10−2 corresponds to a transition from a linear to a weakly nonlinear system dynamics. Periodic boundary conditions have been implemented. The results shown in Figs. 1 and 2 pertain to the linear dynamics regime. It can be shown that in the case, where l = −1 and = −1 Fig. 1 , the characteristic equation Eq. 32 has one real and two complex conjugate roots. From Eq. 34 it is clear that the mode corresponding to the root with negative imaginary part will become dominant for sufficiently late times t. This observation is in full agreement with Fig. 2. In addition, the slope of the asymptotic line matches the imaginary part of the complex roots with an accuracy on the order of 10−4. In contrast, when l = 1 and = 1 the characteristic equation possesses three real roots and these correspond to the three eigenfrequencies of systems 1, 2, and 3. By Fourier analyzing the numerical data for the amplitude of the electrostatic potential a t , the values of these frequencies can be obtained and compared to the roots of the characteristic equation. As Fig. 2 shows the two results are in excellent agreement. Increasing from 10−3 to 10−2 effectively switches on the nonlinearity. It can be shown that phase-matching resonance condition 53 is satisfied at v0 = v pm 0.49. In order to assess the efficiency of the FWM process, Eqs. 54 – 56 have been solved for several values of v0 in the vicinity of v pm. It was found that significant energy exchange between the modes occurs on time scales on the order of 104. Solving the original system of hydrodynamic equations Eqs. 10 , 11 , and 13 on such a time scale is not practical and given the small efficiency of the FWM process it can be neglected. Figures 3 and 4 compare the numerical data for the current velocity v z ; t with the analytical result, given by Eq. 40 . First, the quantity v , k = 2 has been computed by fast-Fourier transforming the quantity v z ; t exp −2iz and this results in locating the peaks of the spectrum dashed line in Figs. 3 and 4 . In order to increase the accuracy “slow” Fourier transform has been performed in the vicinity of the peaks continuous lines . Finally, the “ *” symbols represent the amplitudes of the components at frequencies m + n computed from Eq. 40 . As can be seen, the analytical and numerical results are in very good agreement. Note that besides the six frequency components at frequencies m + n there are two additional peaks at frequencies = 2 . They represent sound waves, solutions of Eqs. 10 , 11 , and 13 with = 0 and harmonic number k = 2 propagating at Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp 032305-6 K. B. Marinov and S. I. Tzenov Phys. Plasmas 18, 032305 2011 10 −4 σ=1, l=1, v0=0.49, a(0)=0.0055, ε=0.01 2ω −2λ 1 2λ 10 |v(ω,k=2)| −5 2ω 3 ω2+ω3 ω1+ω2 10 −6 ω +ω , 2ω 1 3 2 10 −7 −5 −4 −3 −2 −1 0 ω 1 2 3 4 5 FIG. 5. Color online “Numerical:” density distribution x ; t = 600 obtained from the numerical solution of Eqs. 10 , 11 , and 13 ; linear: Eq. 31 with Am = const. The curve marked nonlinear is obtained by taking into account the nonlinear frequency shift of Am Eqs. 54 – 56 . FIG. 3. Color online Fourier transform v , k = 2 k is the harmonic number of the current velocity v z ; t . The dashed line is the spectrum obtained by a fast-Fourier transform. In addition slow Fourier transform is performed in the vicinity of the peaks in order to improve the frequency resolution and the continuous line is the result. The * symbols represent the analytical result, given by Eq. 40 . The values of the eigenfrequencies are 1 = 1.589, 2 = −0.173, and 3 = −1.925. The two components at frequencies −2 and 2 are sound waves harmonic number k = 2 , propagating at speed . analytical solution. This has been verified by calculating the square difference between the numerical and the linear and nonlinear analytical results, respectively. VI. CONCLUDING REMARKS the sound velocity . Since these waves are not in resonance with any of the six components at frequencies 2 m+ n m + n and their amplitudes remain much smaller than A1, A2, and A3, their effect on the evolution of the latter can be neglected. Figure 5 compares the analytical and numerical results for the density x ; t . The curve labeled “linear” is obtained from Eq. 31 with Am = const. The curve with label “nonlinear” in addition takes into account the slow variation of Am, resulting from the nonlinear self- and cross-modulation effects described by Eqs. 54 – 56 . As can be seen taking into account the nonlinear effect improves the accuracy of the x 10 12 10 |v(ω,k=2)| 8 6 4 2 −0.36 −7 σ=1, l=1, v =0.49, a(0)=0.0055, ε=0.01 0 2ω2 ω1+ω3 −0.35 −0.34 ω −0.33 −0.32 FIG. 4. Color online The same as in Fig. 3 but the frequency region near 2 2 and 1 + 3 is shown in more detail. Using a class of waterbag phase space density distributions, which is an exact solutions to the Vlasov equation, we have cast the single-wave model equations to a fluid dynamic form with nonzero thermal velocity. Since the continuity and momentum balance equations comprise an exact hydrodynamic closure, the hydrodynamic representation is fully equivalent to the original system comprising the Vlasov equation and the evolution equation for the single electrostatic field mode. Based on the renormalization group method, a system of coupled nonlinear equations for the slowly varying amplitudes of interacting plasma density waves has been derived. Depending on the solution of the characteristic equation, the system mentioned above takes the form of either three coupled nonlinear equations describing the process of threewave interaction in the case, where all three roots are real, or two coupled equations with complex coefficients if the characteristic equation possesses one real and two complex conjugate roots. In linear dynamic mode and three real roots of the characteristic equation , the values of the three eigenfrequencies 1, 2, and 3 computed numerically have been found in excellent agreement with the solutions of the characteristic equations Eq. 32 . Similarly, in the case of a single real root, the asymptotic behavior of the electrostatic potential is in full accordance with the value of imaginary part of the complex roots. In the weakly nonlinear regime, the spectrum of the nonlinear correction to the current velocity closely matches the analytical solution. Finally, direct comparison between the analytical expressions for the density x ; t , with and without nonlinear effects, and its numerical solution via numerical integration of the hydrodynamic equations Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp 032305-7 Nonlinear density waves in the single-wave model 3 Phys. Plasmas 18, 032305 2011 W. E. Drummond, J. H. Malmberg, T. M. O’Neil, and J. R. Thompson, Phys. Fluids 13, 2422 1970 . 4 T. M. O’Neil, J. H. Winfrey, and J. H. Malmberg, Phys. Fluids 14, 1204 1971 . 5 N. G. Matsiborko, I. N. Onishchenko, V. D. Shapiro, and V. I. Shevchenko, Plasma Phys. 14, 591 1972 . 6 A. Simon and M. Rosenbluth, Phys. Fluids 19, 1567 1976 . 7 P. A. E. M. Janssen and J. J. Rasmussen, Phys. Fluids 24, 268 1981 . 8 J. Denavit, Phys. Fluids 28, 2773 1985 . 9 M.-C. Firpo, F. Doveil, Y. Elskens, P. Bertrand, M. Poleni, and D. Guyomarc’h, Phys. Rev. E 64, 026407 2001 . 10 M.-C. Firpo, F. Leyvraz, and G. Attuel, Phys. Plasmas 13, 122302 2006 . 11 A. Antoniazzi, G. De Ninno, D. Fanelli, A. Guarino, and S. Ruffo, J. Phys.: Conf. Ser. 7, 143 2005 . 12 R. C. Davidson, H. Qin, S. I. Tzenov, and E. A. Startsev, Phys. Rev. ST Accel. Beams 5, 084402 2002 . 13 S. I. Tzenov, Contemporary Accelerator Physics World Scientific, Singapore, 2004 . 14 G. P. Agrawal, Nonlinear Fiber Optics Academic, New York, 2001 . 15 B. Bruhn, Phys. Plasmas 13, 023505 2006 . 16 B.-P. Koch, N. Goepp, and B. Bruhn, Phys. Rev. E 56, 2118 1997 . 17 M.-C. Firpo and Y. Elskens, Phys. Rev. Lett. 84, 3318 2000 . shows that taking the nonlinear effects into account improves the agreement between the analytical and numerical results. It is worthwhile to mention that similar equations have been obtained earlier by Janssen and Rasmussen.7 Although in a different physical context, amplitude equations similar to the ones derived in Sec. V have been reported by Bruhn and co-workers.15,16 Recently, numerical simulations of a discrete self-consistent wave-particle model describing an ensemble of particles coupled to a monochromatic wave have revealed a new phenomenon, a phase transition associated with the Landau damping regime.17 This latter finding adds particular proof to the result presented here. An interesting continuation of the present study would be the consideration of large-scale hydrodynamic fluctuations and their influence on the dynamics of perturbations. This we plan to complete in a future publication. 1 2 D. del-Castillo-Negrete, Phys. Plasmas 5, 3886 1998 . J. D. Crawford and A. Jayaraman, Phys. Plasmas 6, 666 1999 . Author complimentary copy. Redistribution subject to AIP license or copyright, see http://php.aip.org/php/copyright.jsp