Dynamics of a charged particle in a progressive plane wave

Physica D 238 (2009) 226–232

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Dynamics of a charged particle in a progressive plane wave A. Bourdier ∗ Département de physique Théorique et Appliquée, CEA/DAM Ile-de-France, Bruyères-le-Châtel, 91297 Arpajon Cedex, France

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Article history: Received 24 April 2008 Received in revised form 15 August 2008 Accepted 3 September 2008 Available online 25 September 2008 Communicated by J. Garnier

a b s t r a c t The dynamics of a charged particle in a relativistic strong electromagnetic plane wave propagating in a medium is studied. The problem is shown to be integrable when the wave propagates in vacuum. When it propagates in plasma, and when the full plasma response is considered, an exhaustive numerical work allows us to conclude that the problem is not integrable. © 2008 Elsevier B.V. All rights reserved.

PACS: 05.45.-a 52.35.Mw Keywords: Wave propagation in plasma Integrability Chaos

1. Introduction The study of the relativistic dynamics of a charged particle in electromagnetic fields is of prime importance in high intensity laser–matter interaction. Integrability and chaos are the key to predicting conditions for stochastic heating which can take place when very nonlinear regimes are reached. Recently, particle-incell (PIC) code simulations results published by Tajima et al. [1] and also theoretical results obtained by Mulser et al. [2] have shown that the irradiation of very high intensity lasers on clustered matter leads to a very efficient heating of electrons. Tajima et al. [1], and Kanapathipillai [3] have shown that chaos seems to be the origin of the strong laser coupling with clusters. It was confirmed in PIC code simulations, in the case of two counterpropagating laser pulses, that stochastic heating can lead to efficient electron acceleration [4,5]. In this field, many issues need to be studied. The one that we will address below is the stability of the electron motion in the fields of electromagnetic wave propagating in vacuum or in plasma. At very high intensities, the motion of a charged particle in a wave is highly non-linear. The situations when the motion is integrable are exceptional. The solutions corresponding to these situations deserve to be studied as they are strong as predicted by KAM theorem [6–11].

∗

Tel.: +33 169266137; fax: +33 169267106. E-mail addresses: [email protected], [email protected].

0167-2789/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2008.09.004

The main goal of this paper is to explore the influence of the plasma on the stability of electron trajectories. In order to make it clear, we recall a certain number of results that were previously obtained. The dynamic of a charged particle in a linearly, or almost linearly, polarized traveling high intensity wave is studied when it propagates in a nonmagnetized medium. The problem is shown to be Liouville integrable when the wave propagates in vacuum. In this situation, the equation of Hamilton–Jacobi is also solved showing the integrability of the system again and also giving resonance conditions when a perturbing wave is taken into account [5a,e,f], [12]. When the wave propagates in plasma, the plasma response is partly taken into account by introducing an index, n, in the equations. Then, the problem is still integrable. The effect of the full plasma response is studied next by considering plasma wave equations. In their paper, Kaw et al. [13] have shown an apparent absence of chaos for the system which was thus promoted to a good position in the list of systems waiting for the proof of their integrability. An exhaustive numerical study of the system allowed us to witness chaotic trajectories, showing that the system is not integrable. The stochastic behaviour of trajectories was first seen in the laboratory frame by performing Poincaré maps and confirmed by calculating a nonzero Lyapunov exponent. The same kind of numerical work was achieved in a special Lorentz frame where the variables which describe the waves are space independent [14]. The numerical work happens to be easier in this frame as at least some chaotic trajectories fill larger volumes in phase space.

A. Bourdier / Physica D 238 (2009) 226–232

227

Consequently, the generating function defined by Eq. (4) yields the canonical spatial transformation Q¯ i = zˆ − tˆ = ς .

(6)

This canonical transformation keeps Pˆ z unchanged and  yields  ^ = a new normalized Hamiltonian given by H xˆ , Pˆ x , ς , Pˆ z







ˆ xˆ , Pˆx , z , Pˆz , tˆ + ∂/∂ tˆ F2 zˆ , Pˆz , tˆ H ^

H = Cˆ =

2.1. The wave propagates in vacuum 2.1.1. Hamiltonian formulation of the problem—integrability of the system Let us consider a charged particle in an electromagnetic plane wave propagating along the z direction (Fig. 1). The following 4-potential is chosen [φ, A] = 0,

ω0



cos(ω0 t − k0 z )ˆex ,

(1)

where E0, ω0 , k0 are constants and eˆ x is the unit vector along the x-axis. The relativistic Hamiltonian of p a charged particle in an

electromagnetic wave is given by H = (P + eA)2 c 2 + m2 c 4 − eφ , where −e, m, and P are respectively the particle’s charge, its rest mass and its canonical momentum. In the present situation, it reads

# 12 2 eE0 2 2 2 2 2 2 4 Px + cos (ω0 t − k0 z ) c + Py c + Pz c + m c , (2) ω0

" H =

where the Pi (i = x, y, z ) are the canonical momentum components. This system has three degrees of freedom. The following dimensionless variables and parameter are introduced next xˆ = k0 x,

yˆ = k0 y,

tˆ = ω0 t ,

ˆ = H

H mc 2

zˆ = k0 z ,

,

a=

eE 0 mω0 c

Pˆ x,y,z =

Pˆ x + a cos ς

2

,

that is to say

 2 2 ˆ ˆ + Py + Pz + 1 − Pˆz .

(7)

^

2. Dynamics of a charged particle in a linearly polarized electromagnetic traveling wave

E0





This Hamiltonian is autonomous as it is a function of the three momenta Pˆ x , Pˆ y , Pˆ z and ς which is a space coordinate (we have

Fig. 1. Frame of reference.






Px,y,z mc

called the value of the Hamiltonian Cˆ ), H , Pˆ x and Pˆ y are three constants of motion, which are independent and in involution and, as a consequence, the system is completely integrable [6–10]. 2.1.2. Integration of the Hamilton–Jacobi equation When using the proper time of the particle to parameterize the motion in the extended phase space, the Hamiltonian of the charged particle in the wave reads [17] H =

1 2

mc 2 γ 2 −

1 2m

1

(P + eA)2 − mc 2 ,

(8)

2

where γ is the Lorentz factor. Within the scope of this Hamiltonian formulation one has H = 0 [17] This zero-value Hamiltonian is satisfactory in the sense that it allows us to derive the right equations of motion. Then, the dimensionless variables and parameter defined by Eq. (3) are used again, a normalized proper time: τˆ = ω0 τ and a normalized vector potential: a = eA/mc are also introduced. The normalized Hamiltonian reads:

ˆ = H

1 2

γ2 −

2 1 ˆ + a − 1. P 2 2

(9)

It is assumed that the electron motion is restricted to the plane of polarization of the wave (the y degree of freedom is
assumed not to be excited). We look for a set of actions P⊥ , Pk , E and angles (θ , ϕ, φ), instead of the configuration (r, t ) , and  momentum, (P, −γ ) in the

xˆ , zˆ , tˆ, Pˆ x , Pˆ z , −γ

phase space. It

will be shown further that P⊥ and Pk represent the drift motion of the charged particle and E its average energy normalized  to mc 2 . We seek a canonical transformation xˆ , zˆ , tˆ, Pˆ x , Pˆ z , −γ

→

θ , ϕ, φ, P⊥ , Pk , E , such that the new momenta are constants of



, (3)

,

and the canonical transformation, (qi , pi ) → Q¯ i , P¯ i , given by the type-2 generating function

motion. We look for a generating function, Fˆ2 P⊥ , Pk , E , xˆ , zˆ , tˆ , such that Pˆ x = ∂ F2 /∂ xˆ , Pˆ z = ∂ F2 /∂ zˆ , γ = ∂ F2 /∂ t which is a solution of the Hamilton–Jacobi equation [7–10,18,19].








F2 zˆ , Pˆ z , tˆ = Pˆ z zˆ − tˆ ,




(4)

is performed [6–10,15,16]. Let us mention that when
performing a canonical transformation (qi , pi ) → Q¯ i , P¯ i defined by a type-2 generating function F2 qi , P¯ i , t , where P¯ i and Q¯ i are the new coordinates, and pi and qi the old ones, the canonical transformation is given by [6–10]





∂ F2 qi , P¯i , t pi = , ∂ qi
∂ F2 qi , P¯i , t Q¯ i = . ∂ P¯i

ˆ H



xˆ , zˆ , tˆ,

∂ F2 ∂ F2 ∂ F2 , , ∂ xˆ ∂ zˆ ∂ tˆ



= Hˆ 0 = 0.

Eqs. (9) and (10) lead to



∂ F2 ∂ tˆ

2

 −

  
2 ∂ F2 ∂ F2 2 ˆ + a cos t − zˆ − − 1 = 0. ∂ xˆ ∂ zˆ

(11)

Following Landau and Lifshitz [19], it is assumed that F2 has the following form F2 = α1 tˆ + α2 xˆ + α3 zˆ + F tˆ − zˆ ,




(5)

(10)

(12)

where the quantities α1 , α2 and α3 are chosen to be three independent constants of integration in order to select three new

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A. Bourdier / Physica D 238 (2009) 226–232

constant momenta. Letting ζ = tˆ − zˆ and F 0 (ζ ) = ∂ F (ζ ) /∂ζ , Eq. (11) becomes

2.2. The wave propagates in plasma

 2 2  α1 + F 0 (ζ ) − (α2 + a cos ζ )2 − α3 − F 0 (ζ ) = 1.

2.2.1. The plasma index of refraction is introduced to describe a part of the plasma influence A part of the influence of the plasma is taken into account first by simply introducing the index of the medium in the equation of propagation of waves in vacuum. This is done as in the case of a plane wave propagating along a constant homogeneous magnetic field; including an index of refraction in this very simple way is enough to make the system non-integrable [5a,d,f]. The wave vector potential is supposed to be given by Eq. (1). The dimensionless variables and parameters previously defined are used again. The normalized equations can be derived through the ˆ = nγ = nH /mc 2 , where n is the index normalized Hamiltonian: H of refraction of the plasma. This Hamiltonian reads

(13)

Solving this equation leads to F (ζ ) =

α2 a sin ζ α1 + α3   ζ 1 + sin(2ζ ) + F0 ,

1 − α12 + α22 + α32 2(α1 + α3 )

+

a 2(α1 + α3 )

ζ+

2

(14)

4

where F0 is a constant with respect to ζ which is assumed to vanish as only partial derivatives of F (ζ ) are considered. Letting E=−

1 − α12 + α22 + α32

Pk = −

2 (α1 + α3 )

1 − α12 + α22 + α32

P⊥ = α2 ,

2 (α1 + α3 )

− α1 − + α3 −

a 4 (α1 + α3 ) a

,

4 (α1 + α3 )

,

(15)

as: α1 + α3 = Pk − E, the following expression is found for F2 [12,19] Fˆ2 P⊥ , Pk , E , xˆ , zˆ , tˆ = Pk zˆ + P⊥ xˆ − E tˆ

ˆ =n H



Pˆ x + a cos tˆ − zˆ


2

1/2 + Pˆy2 + Pˆz2 + 1 .

(21)

ˆ − Pˆ is also a Pˆ x and Pˆ y are two obvious constants of motion. Cˆ = H constant of motion. As these three constants are independent and in involution, the system is still integrable.




+

P⊥ a Pk − E


sin tˆ − zˆ +

a2 8 Pk − E



sin 2 tˆ − zˆ .

(16)

The old configuration variables expressed in terms of the new ones are given by [12] pˆ x = P⊥ + a cos (φ + ϕ) , a2 P⊥ a
cos 2 (φ + ϕ) , pˆ z = Pk − cos (φ + ϕ) − Pk − E 4 Pk − E 2 P⊥ a a
cos 2 (φ + ϕ) , γ =E− cos (φ + ϕ) − Pk − E 4 Pk − E a (17) xˆ = θ + sin (φ + ϕ) , Pk − E P⊥ a a2 zˆ = ϕ −
2 sin 2 (φ + ϕ) ,
2 sin (φ + ϕ) − 8 Pk − E Pk − E P⊥ a a2 tˆ = −φ −
2 sin 2 (φ + ϕ) ,
2 sin (φ + ϕ) − 8 Pk − E Pk − E

2.2.2. The full plasma response is taken into account through plasma wave equations Following Akhiezer and Polovin [20], the propagation of a relativistically strong wave in a cold relativistic electron plasma is described with Maxwell and Lorentz equations. All the variables entering into these equations are assumed not to be functions of space and time separately, but only of the combination i.r − Vt, where i is a constant unit vector parallel to the direction of propagation of the wave, and V a constant. Then, the equations describing non-linear traveling waves propagating in the absence of the external magnetic field are as follows: d2 pˆ x dτ 2

+ ωp2

β2 β pˆ x p = 0, 2 β − 1 β 1 + pˆ 2 − pˆ z

β pˆ y β2 p = 0, dτ 2 β 2 − 1 β 1 + pˆ 2 − pˆ z  p d2  β 2 pˆ z 2 + ω2 p ˆ β p − 1 + p = 0, z p dτ 2 β 1 + pˆ 2 − pˆ z d2 pˆ y

+ ωp2

(22a)

(22b)

(22c)

where pˆ x and pˆ z are the components of the normalized momentum of the charged particle (ˆpi = pi /mc ). It is straightforward to check  that, P⊥ = pˆ x , Pk = pˆ z , E = hγ i (hi stands for averaging

where pˆ = p/mc is the normalized hydrodynamic momentum, τ = t − i.r/V , β = V /c, and ωp is the plasma frequency. Letting

over the oscillations) and Cˆ = E − Pk . All the former constants of motion are found again. Pk and P⊥ describe the uniform motion of translation which is obtained by averaging out the oscillatory part of the motion. The new Hamiltonian in terms of the action variables reads [12]

d2 pˆ x

˜ Pk , P⊥ , E = − H


1 2

2 M 2 + Pk2 + P⊥ −E


2

,

(18)

˜ = 0 [17], the energy momentum where M 2 = 1 + a2 /2. As H dispersion relation is given by E Pk , P⊥ , a =




q

2 M 2 + Pk2 + P⊥ .

(19)

The solution of Hamilton equations is

θ = −P⊥ τˆ ,

ϕ = −Pk τˆ ,

φ = E τˆ .


−1/2 θ = ωp β 2 − 1 τ , these equations read β 3 pˆ x p + = 0, dθ 2 β 1 + pˆ 2 − pˆ z

β 3 pˆ y p + = 0, dθ 2 β 1 + pˆ 2 − pˆ z
 p β 2 β 2 − 1 pˆ z d2  2 β pz − 1 + pˆ + p = 0. dθ 2 β 1 + pˆ 2 − pˆ z d2 pˆ y

(23a)

(23b)

(23c)

When consideringthat the electron motion is in the x–z plane 

(pˆ y = 0), we let X =

p

p β 2 − 1 pˆ x and Z = β pˆ z − 1 + pˆ 2 , then,

Eq. (23) become: (20)

Finding the solution of Hamilton–Jacobi equation is another way to prove that the problem is integrable. This solution is also very useful to predict resonances when a perturbing mode is considered [5a,e,f], [12].

d2 X dθ

2

d2 Z dθ

2

+p +p

β 3X β2 − 1 + X 2 + Z2 β 3Z

β2 − 1 + X 2 + Z2

= 0,

(24a)

+ β 2 = 0.

(24b)

A. Bourdier / Physica D 238 (2009) 226–232

229

Fig. 2. Poincaré surface of section plots in the laboratory frame: PZ versus Z (X = 0, Px > 0). H = 20, β = 1.2. (a), (b): Several trajectories are considered. (c): Only one trajectory is considered. H = 50, β = 1.2. (d): Several trajectories are considered. (e): Only one trajectory is considered.

Eq. (24) can obviously be derived from the Hamiltonian 1 2 PX + PZ + β 3 β 2 − 1 + X 2 + Z 2 + β 2 Z , (25) 2 where PX = dX /dθ and PZ = dZ /dθ . Then, the coupled stationary wave problem can be treated as a two-degrees-offreedom conservative problem. Consequently, Poincaré maps can be performed. Since the Hamiltonian is symmetric in X , the X oscillator can be used as a clock and PZ versus Z can be plotted each time X = 0 and PX > 0. The problem looks integrable; most of the Poincaré maps obtained are regular (Fig. 2) and most of the Lyapunov exponents calculated go to zero. Still, chaos was found considering one trajectory for which chaos takes place in a very small volume in phase space (Fig. 3). Chaos appears when enlarging a lot the place where it takes place, therefore, it was difficult to know if it were numerical chaos or real chaos (Fig. 3b). Consequently, the lyapunov exponent of this trajectory was carefully calculated using Benettin’s method [6,10, 21]. To do this, two very close trajectories are considered, the H =


2

p

very small distance between them being initially d0 . A sequence dn corresponding to these trajectories is calculated numerically. For every fixed time ∆t, or for every fixed distance ratio d/d0 , dn is renormalized to d0 . The two ways to renormalize are used and compared. Fig. 4 shows the good agreement obtained for the nonzero the Lyapunov exponent calculated when using the two renormalization techniques. It shows that it is a real chaos and that, consequently, the problem is not integrable. Introducing θ 0 = β 3 θ , n = 1/β , Eq. (24) become d2 X dθ 0 2 d2 Z dθ 0 2

+p +p

X

β2 − 1 + X 2 + Z2 Z

β − 1 + X2 + Z2 2

= 0,

(26a)

+ n = 0,

(26b)

Letting, P˜ X = dX /dθ 0 and P˜ Z = dZ /dθ 0 , dropping the tildes for the sake of simplicity, Eq. (26) can be derived from the following

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A. Bourdier / Physica D 238 (2009) 226–232

Fig. 4. Lyapunov exponents: (a) Lyapunov exponents calculated for the same initial conditions as those of the trajectory shown in Fig. 3a, b. The two renormalization techniques are performed, they give almost the same results. (b) Lyapunov exponent corresponding to a regular trajectory.

Fig. 3. Poincaré surface of section plots in the laboratory frame: PZ versus Z (X = 0, Px > 0). H = 248.824, β = 3.24. (a): One trajectory is considered. (b) Enlargement of part of the stochastic orbit.

Hamiltonian H =

1 2

Px2 + Pz2 +




p

β 2 − 1 + X 2 + Z 2 + nZ .

(27)

Then, the problem is under the form studied by Grammaticos et al. [22]. In their paper, they claim that this system is not integrable. Following their paper very high energies are considered (X 2 + Z 2  β 2 − 1), then Eq. (26) read d2 X dθ

02

d2 Z dθ

02

+√

X

X2 + Z2

+√

Z

X2

+ Z2

= 0,

(28a)

+ n = 0.

(28b)

These equations can be derived from the Hamiltonian H =

1 2

Px2 + Pz2 + V (X , Z ) ,




(29a)

with V (X , Z ) =

p

X 2 + Z 2 + nZ .

(29b)

This system corresponds to the limit of infinite energy. Hamiltonian (29) is Hamiltonian (27) when β 2 = 1. It was shown that if Hamiltonian (27) were meromorphic, then system (27) would not be integrable when system (29) is not [23]. It is not clear a priori

Fig. 5. Poincaré surface of section plots in the laboratory frame: PZ versus Z (X = 0, Px > 0). H = 248.824, β = 3.24. One trajectory is considered.

whether this property can be applied to our problem as Hamiltonian (27) is not meromorphic. Still, the right system is described by this Hamiltonian (Eq. (29)) plus a small quantity which depends on β (in the limit of high energies). Then, if system (29) is not integrable, system (27) is probably not integrable for an arbitrary value of this parameter; in other words, the exact system which corresponds to an arbitrary energy is likely to be nonintegrable. Poincaré maps have been performed within this high energy limit system. They show that the system defined by Hamiltonian (29) exhibit chaotic trajectories in some conditions (Fig. 5). This backs up the previous assertion which says that the exact system is not integrable. The Lyapunov exponent for this trajectory is calculated by using Benettin’s method [6,10,21]. The two ways to renormalize are used and compared; Fig. 6 shows the good agreement obtained for the nonzero Lyapunov exponent when using the two renormalization techniques. Then, another frame is considered to look for chaotic trajectories. This is to make absolutely sure that this very important result concerning the integrability of the problem is right. Following Winkles and Eldridge [14] and Romeiras [24], a new frame (L∗ ) which

A. Bourdier / Physica D 238 (2009) 226–232

Px0 = Px ,

231 0

Py0 = Py ,

Pz = Γ



U

Pz −

c2

 E

,

(30)

0

E = Γ (E − UPz ) ,


−1/2

where Γ = 1 − U 2 /c 2 and E = γ mc 2 is the energy of the charged particle. In the extended phase space, where time is treated on a common basis with other coordinates, a fully covariant Hamiltonian formulation of the problem can be constructed. In this space, the Lorentz transformation defined above is identical to the canonical transformation generated by the following type-2 generating function F2 x, y, z , t , Px0 , Py0 , Pz0 , E 0 = Px0 x + Py0 y + Γ






Pz0 +


− Γ E 0 + UPz0 t .

Fig. 6. Lyapunov exponents: (a) Lyapunov exponents calculated for the same initial conditions as those of the trajectory shown in Fig. 5. The two renormalizations are performed; they give almost the same results. (b) Lyapunov exponent corresponding to a regular trajectory.

moves uniformly along the z-axis with velocity U relative to the laboratory frame is introduced. The Lorentz transformation of the 4-momentum is given by [17,19].

U 0 E c2

 z (31)

As a consequence, if the hydrodynamic motion is not integrable in the frame (L∗ ), it is also not integrable in the laboratory frame. The phase of the wave which is an invariant takes, in the moving frame, the following form [17,19]



ω0 t − k0 z = Γ ω0



U t + 2 z0 c 0



0

− k0 z + Ut

0




.

(32)

When the phase velocity of the wave is greater than the speed of light, there exists one special frame (L∗ ) in which the phase does not depend on the variable z 0 . This frame has the following

Fig. 7. Poincaré surface of section plots in (L∗ ) : PZ versus Z (X = 0, Px > 0). H = 10, n = 0.3. (a): One trajectory is considered. (b) Enlargement of part of the stochastic orbit H = 20, n = 0.3. (c): One trajectory is considered. (d) Enlargement of part of the stochastic orbit.

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A. Bourdier / Physica D 238 (2009) 226–232

account by considering plasma wave equations. Until now, the apparent absence of chaos for the system would make it hold pending for the proof of its integrability. Here, an exhaustive and cumbersome numerical work allowed us to see chaotic trajectories in the laboratory frame, showing that the system is not integrable. The same kind of numerical work was achieved in a special Lorentz frame where the variables which describe the waves are space independent. In this frame, at least some chaotic trajectories, fill large volumes in phase space. One very important consequence of the nonintegrability of the plasma wave equations is that the Hamilton–Jacobi equation cannot be solved. This implies that, when introducing a perturbing wave in order to generate stochastic heating, one cannot use a classical perturbation method to predict resonances and apply Chirikov criterion. Acknowledgement Fig. 8. Lyapunov exponents: (a) Lyapunov exponents calculated for the same initial conditions as those of the trajectory shown in Fig. 4c, d. The two renormalizations techniques are performed and give almost the same results. (b) Lyapunov exponent corresponding to a regular trajectory.

The author wishes to thank Dr. S. Bouquet for very valuable discussions. References

drift velocity: U /c = k0 c /ω0 = n [14]. Still considering that the electron motion is in the x–z plane (ˆpy = 0), the wave equations (23) read d2 pˆ 0x dθ 0 2 d2 pˆ 0z dθ

02

+p +

pˆ 0x 1 + pˆ 2

= 0,

(33a)

!

pˆ 0z

p

1 + pˆ 2

+ n = 0.

(33b)

where θ 0 = ωp0 t 0 [ωp0 is the plasma frequency in (L∗ )]. Introducing X 0 = pˆ 0x , Z 0 = pˆ 0z , PX0 0 = dX 0 /dθ 0 and PZ0 0 = dZ 0 /dθ 0 , dropping the primes for convenience, Eq. (33) can be derived from the following Hamiltonian H =

1 2

Px2 + Pz2 +




p

1 + X 2 + Z 2 + nZ .

(34)

Then, the problem is under the form studied by Grammaticos et al. again [22]. Poincaré maps are performed, PZ versus Z is plotted each time X = 0 and PX > 0. Fig. 7 show chaotic trajectories. In one case chaos is confirmed by calculating a nonzero Lyapunov exponent (Fig. 8). In short, it has been shown numerically that in both the laboratory frame and in (L∗), the problem is not integrable. 3. Conclusion The dynamic of a charged particle in a linearly or almost linearly polarized traveling high intensity wave which propagates in a medium has been studied. The problem was shown to be integrable when the wave propagates in vacuum. When the wave propagates in plasma, the full plasma response was taken into

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