Using JMP format

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 40, NUMBER 8

AUGUST 1999

Time-asymptotic traveling-wave solutions to the nonlinear Vlasov–Poisson–Ampe`re equations Carlo Lancellotti University of Virginia, Charlottesville, Virginia 22903-2442

J. J. Dorninga) Mathe´matiques pour l’Industrie et la Physique (UMR CNRS 5640), Universite´ Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France

共Received 12 January 1999; accepted for publication 28 April 1999兲 We consider the Vlasov–Poisson–Ampe`re system of equations, and we seek solutions for the electric field E(x,t) that are periodic in space and asymptotically almost periodic in time. Introducing the representation E(x,t)⫽T(x,t)⫹A(x,t) 共where T and A are, respectively, the transient and time-asymptotic parts of E兲 enables us to decompose the nonlinear Poisson equation into a transient equation and a time-asymptotic equation. We then study the latter in isolation as a bifurcation problem for A with the initial condition and T as parameters. We show that the Fre´chet derivative at a generic bifurcation point has a nontrivial null space determined by the roots of a Vlasov dispersion relation. Hence, the bifurcation analysis leads to a general solution for A given 共at leading order兲 by a discrete superposition of traveling-wave modes, whose frequencies and wave numbers satisfy the Vlasov dispersion relation, and whose amplitudes satisfy a system of nonlinear algebraic equations. In applications, there is usually a finite number of roots to the dispersion relation, and the equations for the time-asymptotic wave amplitudes reduce to a finite dimensional bifurcation problem in terms of the amplitude of the initial condition. © 1999 American Institute of Physics. 关S0022-2488共99兲01208-6兴

I. INTRODUCTION

The subject of this work is the well-known Vlasov–Poisson–Ampe`re 共VPA兲 system of equations

⳵f␣ ⳵ f ␣ q␣ ⳵ f ␣ ⫹v ⫹ E ⫽0, ⳵t ⳵x m␣ ⳵v ⳵E ⫽4 ␲ ⳵x

共1a兲

兺␣ q ␣ 冕 d v f ␣ ,

共1b兲

兺␣ q ␣ 冕 d vv f ␣ ,

共1c兲

⳵E ⫽⫺4 ␲ ⳵t

for the propagation of longitudinal electric signals in a collisionless plasma. Here, E(x,t) is the macroscopic longitudinal electric field, and ␣ ⫽1,..., N S are the indexes of the N S different species of particles present in the plasma, each with charge q ␣ , mass m ␣ and single-particle distribution function f ␣ (x, v ,t). A large body of recent mathematical literature1–4 has produced extensive results on existence, uniqueness, and regularity of both classical and weak solutions to these equations. However, any concrete calculation of these solutions is made very difficult, especially in the long-time limit, by the strong nonlinearity of the problem. a兲

Permanent address: University of Virginia, Charlottesville, Virginia 22903-2442.

0022-2488/99/40(8)/3895/23/$15.00

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© 1999 American Institute of Physics

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3896

J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

Physically, the nonlinearity in the VPA system corresponds to the self-consistent feedback from the electric field on the distribution functions. Interestingly, in many wave-propagation problems we expect this feedback to become less and less important in the long-time limit, because the anharmonic mixing of the single-particle trajectories makes the plasma less and less able to exchange energy with the electric field in any coherent fashion. This fact was first pointed out in the physics literature by O’Neil.5 In a classic paper, he argued that a sinusoidal perturbation to a thermal equilibrium will, in general, either Landau damp to a zero electric field6 or lead to a nonzero time-asymptotic solution for the electric field, depending on the amplitude of the initial disturbance and on the magnitude of the Landau damping rate. O’Neil assumed the nonzero time-asymptotic solution for E to be given by a single traveling wave of constant amplitude, but in more realistic physical situations one can reasonably conjecture that the final state will be comprised of more than one wave mode. In fact, this scenario has been confirmed by a significant amount of experimental and numerical evidence.7–9 Recent rigorous investigations10–12 have shown that the nonlinear VPA system does indeed admit undamped traveling-wave solutions like those suggested in the physics literature. Particularly relevant to what will follow are the nonlinearly superimposed traveling-wave BGK-type13 solutions obtained by Buchanan and Dorning.12 However, it is not known whether any of these solutions can be reached as time-asymptotic limits of solutions to the initial value problem for the VPA system. In this article we develop a new procedure for the analysis of the long-time behavior of the solutions to the VPA system for a certain class of initial conditions. Our method is based on the representation of the electric field as the sum of a transient term and a time-asymptotic term, and correspondingly, on the decomposition of the VPA problem into a transient part and a timeasymptotic part. This decomposition turns out to be fruitful in a number of ways, not all of which will be explored in this paper. For instance, we shall not analyze the transient problem, which seems amenable to a relatively straightforward perturbation analysis14 since most of the wellknown secularities in the VPA problem are associated with time-asymptotic evolution. We shall focus, instead, on the time-asymptotic part of the problem; the basic idea will be to show that this time-asymptotic part can be studied in isolation as a bifurcation problem for the time-asymptotic electric field with the initial condition and the transient field playing the role of parameters. Our main result shows that if the VPA system possesses a nonzero small-amplitude time-asymptotic solution 共in a sense that will be defined below兲, then the corresponding electric field is given at leading order by a superposition of traveling-wave modes associated with the roots of a ‘‘timeasymptotic’’ Vlasov dispersion relation. This dispersion relation is completely determined by the initial condition and by the transient electric field, and the same is true for the amplitudes of these traveling-wave modes, which satisfy a nonlinear system of algebraic equations. II. PRELIMINARIES

Throughout this study E and f ␣ will be assumed to be bounded C1 functions of their arguments, with f ␣ ⭓0. In fact, it is enough to assume f ␣ to be bounded and non-negative at time zero, since then the Vlasov equation implies the a priori bounds 0⭐ f ␣ 共 x, v ,t 兲 ⭐ sup f ␣ 共 x, v ,0兲 ⬅M ␣ . 共 x, v 兲

共2兲

We also shall assume f ␣ and v f ␣ to be integrable functions of v on R so that the charge and current densities are well defined. Both E and the f ␣ will be spatially 2␲-periodic. In this case it is easy to show that replacing the Ampe`re equation by its spatially averaged form dE 0 ⫽⫺2 dt

兺␣ q ␣ 冕⫺ ␲ dx 冕 d vv f ␣ ⫹␲

共1c⬘兲

共where E 0 (t) is the k⫽0 spatial Fourier component of E(x,t)兲 yields a system of equations which is completely equivalent to Eqs. 共1兲. Hence, we shall write the VPA system in the compact form

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

3897

⳵f␣ ⳵ f ␣ q␣ ⳵ f ␣ ⫹v ⫹ E ⫽0, ⳵t ⳵x m␣ ⳵v

共3a兲

E⫽⌳ 共 f 1 ,..., f N S 兲 ,

共3b兲

where ⌳ is defined by its spatial Fourier components

⌳ k 共 f 1 ,..., f N S 兲 ⬅

冦

4␲ ik

兺␣ q ␣ 冕 d v f ␣ ,k

E 0 共 0 兲 ⫺4 ␲

兺␣ q ␣ 冕0 d ␶ 冕 d vv f ␣ ,0

k⫽0 共4兲

t

k⫽0

which have been obtained from Eq. 共1b兲 and 共1c⬘兲 for k⫽0 and k⫽0, respectively. Here, the f ␣ ,k are the spatial Fourier components of f ␣ , and E 0 (0) is an assigned initial condition for E 0 (t). The spatially uniform part of the Poisson equation, Eq. 共1b兲, reduces to the zero-charge condition

兺␣ q ␣ 冕⫺ ␲ dx 冕 d v f ␣ ⫽0. ⫹␲

共5兲

From the Vlasov equation it follows immediately that this condition is satisfied as long as it holds at t⫽0. The distribution functions satisfy initial conditions of the form f ␣ 共 x, v ,0兲 ⫽F␣ 共 x, v 兲 ⬅F ␣ 共 v 兲 ⫹h ␣ 共 x, v 兲 ,

共6兲

where the F ␣ ( v ) correspond to a Vlasov equilibrium and satisfy the conditions

兺␣ q ␣ 冕 F ␣共 v 兲 d v ⫽0,

共7兲

兺␣ q ␣ 冕 v F ␣共 v 兲 d v ⫽0.

共8兲

q⫽

j⫽

The function h ␣ in Eq. 共6兲 will be taken to have no spatially uniform part, so that

冕

⫹␲

⫺␲

dxh ␣ 共 x, v 兲 ⫽0.

共9兲

Equations 共7兲 and 共9兲 ensure that the f ␣ (x, v ,0), and thus the f ␣ (x, v ,t), satisfy Eq. 共5兲. Clearly, once the f ␣ (x, v ,0) are chosen, all the initial Fourier components for the field E k (0) with k⫽0 are automatically assigned via the Poisson equation. The characteristic system for the Vlasov equation, Eq. 共3a兲, is given by Newton’s equations (dx/dt)⫽ v , (d v /dt)⫽(q ␣ /m ␣ )E(x,t). All the electric fields E appearing in this paper will be such that these equations have global classical solutions that can be extended indefinitely in t according to classic theorems on ODEs.15 Then, the general solution to the Vlasov equation can be written as f ␣ 共 x, v ,t 兲 ⫽F␣ 共 x E0 共 x, v ,t 兲 , v E0 共 x, v ,t 兲兲 ⬅f␣ 共 E,F␣ 兲 ,

共10兲

where we have introduced the the ‘‘inverse trajectories’’ x E0 (x, v ,t), v E0 (x, v ,t) determined by Newton’s equations; these functions associate with each phase-point (x, v ) the initial condition

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J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

(x E0 , v E0 ) at time zero that leads to (x, v ) at time t. We shall use the notation f ␣ (x, v ,t) ⫽f␣ (E,F␣ ) whenever we want to emphasize the functional dependence of the distribution function on E and the initial condition. Similarly, we shall write f␣⬘ 共 E,F␣ 兲 ⫽

⳵ 关 F 共 x E 共 x, v ,t 兲 , v E0 共 x, v ,t 兲兲兴 . ⳵v ␣ 0

共11兲

Substituting Eq. 共10兲 into Eq. 共3b兲 reduces the problem to a single nonlinear equation for E E⫽N共 E,F ␣ ,h a 兲 ,

共12兲

N共 E,F ␣ ,h ␣ 兲 ⬅⌳ 共 f1 共 E,F1 兲 ,...,fN S 共 E,FN S 兲兲 .

共13兲

where we have defined

Equation 共12兲 will be called the VPA equation; here, of course, the Vlasov equation has become part of the definition of N 共through Eqs. 共10兲 and 共4兲兲.

III. A – T DECOMPOSITION

As mentioned in our introductory discussion, we shall seek solutions for E that are the sum of a transient part and a time-asymptotic part, of the form E 共 x,t 兲 ⫽T 共 x,t 兲 ⫹A 共 x,t 兲 .

共14兲

Since our interest is in periodic traveling-wave solutions, all the functions involved will be assumed to be continuously differentiable and periodic in x. As far as the time variable is concerned, a very general class of functions that have the representation in Eq. 共14兲 is given by the asymptotically almost periodic continuous 共a.a.p.c.兲 functions of t.16 Let AP be the set of all the almost periodic continuous 共a.p.c.兲 functions of t uniformly with respect to x.17 Let T be the set of all the continuous functions of t on R⫹ g(x,t) such that limt˜⫹⬁ g(x,t)⫽0 uniformly in x. The space of the a.a.p.c. functions of t on R⫹ 共uniformly with respect to x, periodic and continuously differentiable in x兲 is given by the direct sum AP⫹T. Here, we shall focus on the subspace W傺AP ⫹T of the functions that are also continuously differentiable in t, i.e., W⬅ 兵 W苸C1 共 R⫻R⫹ 兲 s.t. W⫽A⫹T,

A苸AP,

T苸T 其 .

共15兲

As shown by Fre´chet,16 a.a.p.c. functions enjoy many of the classic properties of almost periodic functions. In particular, if W(x,t) is a.a.p.c. in t then it is bounded and uniformly continuous on R⫹ ; in fact, AP, T and W can be made into Banach spaces with the supnorm 储 W 储 ⫽ sup 兩 W 兩 ,

共16兲

I,R⫹

where I⬅ 关 ⫺ ␲ ,⫹ ␲ 兴 . Moreover, the mean value of W, 1 ␴ ˜⬁ ␴

Mt 关 W 共 x,t 兲兴 ⫽ lim

冕

␴

0

W 共 x,t 兲 dt

共17兲

is always well defined and coincides with the mean value of the a.p.c. part of W, as follows from the fact that Mt 关 T(x,t) 兴 ⫽0 ᭙T苸T 共this will be referred to as ‘‘Fre´chet’s Lemma’’ 16 in what follows兲. The existence of the mean is very important because it makes it possible to associate with each a.a.p.c. W its Fourier–Bohr coefficients,

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

1 ␴ ␴ ˜⬁

w 共 ␭,x 兲 ⫽Mt 关 W 共 x,t 兲 e ⫺i␭t 兴 ⫽ lim

冕

␴

0

W 共 x,t 兲 e ⫺i␭t dt

3899

共18兲

which, of course, coincide with the Fourier–Bohr coefficients of the a.p.c. part of W. For this latter part we know that there exist at most a countably infinite set of real numbers ␭ i such that w(␭,x)⬅0 ᭙␭⫽␭ i . 17 Hence, with each a.a.p.c. function W there is associated a unique Fourier– Bohr series W 共 x,t 兲 ⬃

兺 w i共 x 兲 e i␭ t i

共19兲

共here w i (x)⬅w(␭ i ,x)兲. The series in Eq. 共19兲 coincides with the Fourier–Bohr series for the a.p.c. 共time-asymptotic兲 part of W and determines it uniquely. Of course, it does not determine the whole function W, whose transient part has been averaged away in the computation of the coefficients w i . In fact, the Fourier–Bohr series of an a.a.p.c. function gives an explicit representation of the fundamental projection operator P a :W˜AP such that P a E⫽A.

共20兲

By associating with any E苸W its a.p.c. part A, P a will play a major role in what follows, enabling us to ‘‘sort out’’ the essential features of long-time wave propagation from less interesting 共and more complicated兲 transient phenomena. Of course, the validity of the assumption that E(x,t)苸W is far from obvious, since there seems to be no easy way to prove rigorously a priori that the problem admits an asymptotic a.p.c. state. On the contrary, it could be argued that the intricacies of the nonlinear particle dynamics will always generate some ‘‘noise,’’ which cannot be reasonably expected to be a.a.p.c. in time. We are thinking here of the particles belonging to the thin stochastic layers generated by multiwave resonances in the phase plane, as described by Rechester and Stix18 and by Buchanan and Dorning.12 On the other hand, Buchanan and Dorning12 obtained solutions that are a.p.c. to leading order in the field amplitude, and showed that the noise coming from the stochastic layers vanishes exponentially with that amplitude. Motivated by these results, we study the evolution of initial conditions that produce a small time-asymptotic state, and show that the approximate solution for E(x,t) is a.a.p.c. in time, with an error that is negligible with respect to the timeasymptotic field amplitude. Let us substitute the a.a.p.c. representation of E, Eq. 共14兲, into the nonlinear VPA equation, Eq. 共12兲; in order to ensure that this equation is well defined, we need to assume that the integral in dt in Eq. 共4兲 is bounded 共and thus a.a.p.c. 共Ref. 17兲兲 in time. This implies the necessary condition 4␲

冕 d ␶ 冕 d vv f ␣,0共 v , ␶ 兲⫽0 兺␣ q ␣ ␴lim ˜⬁ ␴ 0 1

␴

共21兲

which can also be obtained directly by taking the time-average of the Ampe`re equation, Eq. 共1c兲. We now apply the projection operator P a introduced above to both sides of Eq. 共12兲 in order to decompose the problem into its transient and time-asymptotic components. This yields the following system of two coupled equations for the time-asymptotic field A and the transient field T: A⫽ P ␣ N共 A⫹T,F ␣ ,h ␣ 兲 ,

共22a兲

T⫽ 共 I⫺ P a 兲 N共 A⫹T,F ␣ ,h ␣ 兲 .

共22b兲

Our strategy will be to focus on the ‘‘asymptotic equation,’’ Eq. 共22a兲. We shall show that it is possible to obtain important information about the structure of solutions to Eq. 共22a兲 independently of the details of T. This will be done in the following steps:

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J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

共1兲 First, we shall show that the asymptotic equation can be studied as an infinite-dimensional bifurcation problem, because it possesses a manifold of vanishing solutions corresponding to different choices of the initial perturbation h ␣ and of the corresponding transient field T. 共2兲 Then, we shall show that there is a natural way to linearize Eq. 共22a兲, which leads to a time-asymptotic linear equation, whose solutions are very different from those found in the traditional linear theory. In particular, the time-asymptotic linear theory will yield results that are perfectly consistent with the undamped nonlinear multiple traveling-wave solutions recently discovered by Buchanan and Dorning.12 共3兲 Finally, we shall exploit the properties of the time-asymptotic linear operator in order to reduce the original nonlinear problem to a lower-dimensional system of bifurcation equations for the amplitudes of the leading-order Fourier–Bohr coefficients of A. IV. PURELY TRANSIENT ANALYSIS

We want to show that the time-asymptotic equation, Eq. 共22a兲, is satisfied by the zero timeasymptotic field A⬅0 independently of the choice of the initial condition. Substituting A⬅0 into Eq. 共22a兲 gives the equation P a N共 T,F ␣ ,h ␣ 兲 ⫽0,

共23兲

which in turn entails the analysis of the Vlasov equation for a purely transient field

⳵f␣ ⳵ f ␣ q␣ ⳵ f ␣ ⫹v ⫹ T ⫽0. ⳵t ⳵x m␣ ⳵v

共24兲

Of course, the general solution of this equation has the form f ␣T (x, v ,t) ⫽F␣ (x T0 (x, v ,t), v T0 (x, v ,t)), where the 关 x T0 (x, v ,t), v T0 (x, v ,t) 兴 give the ‘‘inverse particle trajectory,’’ i.e., the starting point at time zero for a particle that arrives at the point (x, v ) at time t under the influence of the electric field T. The fundamental observation is that, as the transient T dies away, it acts on the particles more and more weakly, so that the trajectories tend asymptotically to straight lines as t˜⬁. This suggests that in the long-time limit the distribution function must approach some kind of Vlasov equilibrium, at least in a coarse-grained sense, so that Eq. 共23兲 is satisfied. To establish this, let us consider Newton’s equations for the purely transient, spatially periodic field T: x˙ ⫽ v , v˙ ⫽T(x,t) 共where we have set q ␣ /m ␣ ⫽1 to avoid excessive notational detail兲. For the initial value problem with the initial conditions x(0)⫽x 0 , v (0)⫽ v 0 , these equations can be written in the integral form x 共 t 兲 ⫺x 0 ⫽ v 0 t⫹

冕 ␶冕

v共 t 兲 ⫺ v 0 ⫽

t

␶

d

0

0

冕

t

0

d ␶ ⬘T共 x共 ␶ ⬘ 兲, ␶ ⬘ 兲,

d ␶ T 共 x, 共 ␶ 兲 , ␶ 兲 .

共25a兲

共25b兲

We want the field T(x,t) to decay fast enough, as t˜⬁, that both supx 兩 T(x,t) 兩 and 兰 ⬁t d ␶ ⬘ supx 兩 T(x, ␶ ⬘ ) 兩 be integrable over the positive real axis. This will be ensured when supx 兩 T(x,t) 兩 tends to zero at least as fast as t ⫺ ␩ as t˜⬁, with ␩ ⬎2. Under this condition, it follows immediately from Eq. 共25b兲 that lim v共 t 兲 ⫽ v 0 ⫹ t˜⬁

冕

⬁

0

d␶ T共 x共 ␶ 兲,␶ 兲,

共26兲

where the integral on the right-hand side is a function only of the initial point (x 0 , v 0 ) of the trajectory x(t) in phase space. Hence, we shall write Eq. 共26兲 in the form

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

lim v共 t 兲 ⫽ v 0 ⫹H 共 x 0 , v 0 兲 ⬅ v ⬁ .

3901

共27兲

t˜⬁

Then, Eq. 共25a兲 can be rewritten immediately as x 共 t 兲 ⫺x 0 ⫽ v ⬁ t⫹

冕 ␶冕 t

d

⬁

␶

0

d ␶ ⬘T共 x共 ␶ ⬘ 兲, ␶ ⬘ 兲.

共28兲

Thus, under the above conditions on T, lim 关 x 共 t 兲 ⫺ v ⬁ t 兴 ⫽x 0 ⫹ t˜⬁

冕 ␶冕 ⬁

d

⬁

␶

0

d ␶ ⬘T共 x共 ␶ ⬘ 兲␶ ⬘ 兲

共29兲

which will be written in the form lim 关 x 共 t 兲 ⫺ v ⬁ t 兴 ⫽x 0 ⫹G 共 x 0 , v 0 兲 ⬅x ⬁ .

共30兲

t˜⬁

Equations 共27兲 and 共30兲 express the fact that, as t goes to infinity and T(x,t) tends to zero, each trajectory tends asymptotically to the straight line trajectory starting at t⫽0 from the ‘‘fictitious’’ phase point (x ⬁ , v ⬁ ). That is, 关 x 共 t 兲 , v共 t 兲兴 ——˜ 关 x ⬁ ⫹ v ⬁ t, v ⬁ 兴 .

共31兲

t˜⬁

Now, we want to obtain an analogous result for the ‘‘inverse’’ trajectories 关 x T (x, v ,t), v T (x, v ,t) 兴 . This requires a slightly more sophisticated analysis, which leads to the following theorem: Theorem 1: If there is a number ␩ ⬎2 such that both supx 兩 T(x,t) 兩 and supx 兩 (dT/dx)(x,t) 兩 go to zero as t ⫺ ␩ as t˜⬁, then the corresponding inverse trajectories can be written in the form, ˜ 共 x⫺ v t, v 兲 ⫹ ␶ 1 共 x, v ,t 兲 , x T0 共 x, v ,t 兲 ⫽x⫺ v t⫹G

共32a兲

˜ 共 x⫺ v t, v 兲 ⫹ ␶ 2 共 x, v ,t 兲 , v T0 共 x, v ,t 兲 ⫽ v ⫹H

共32b兲

˜ ,H ˜ 苸C(I⫻R⫹ ) and limt˜⬁ ␶ 1 (x, v ,t)⫽limt˜⬁ ␶ 2 (x, v ,t)⫽0, uniformly in x and v . where G Proof: The proof of this result is based on the fact that, by definition, x T0 (x, v ,t) and v T0 (x, v ,t) must satisfy the Vlasov equations

⳵ x T0 ⳵ x T0 ⳵ x T0 ⫽0, ⫹v ⫹T ⳵t ⳵x ⳵v

共33兲

⳵ v T0 ⳵ v T0 ⳵ v T0 ⫹v ⫹T ⫽0, ⳵t ⳵x ⳵v

共34兲

which can be written in the integral form x T0 共 x, v ,t 兲 ⫽x⫺ v t⫺

v T0 共 x, v ,t 兲 ⫽ v ⫺

冕 ␶ 冋 ⳵⳵ 册 t

d

T

x T0

0

v

冕 冋 册 ⳵ v T0 d␶ T ⳵v 0

,

共35兲

共 x⫺ v共 t⫺ ␶ 兲 , v 兲

t

.

共36兲

共 x⫺ v共 t⫺ ␶ 兲 , v 兲

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J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

In this case, it is not immediately obvious that one can take t˜⬁ in the limits of integration, as was done for Eqs. 共25兲. The core of the proof will be precisely the determination of appropriate bounds on the growth in time of ( ⳵ x T0 / ⳵ v ) and ( ⳵ v T0 / ⳵ v ), in order to show that the integrands on the right-hand sides of Eqs. 共35兲 and 共36兲 are, indeed, integrable at infinity. The superscript ‘‘T’’ will be dropped for the remainder of this proof. Let us consider, first, the ‘‘direct’’ trajectories, written in the form x(x 0 , v 0 ,t), v (x 0 , v 0 ,t). We shall exploit the decay properties of T in order to obtain bounds on the growth in time of the partial derivatives ⳵ x/ ⳵ x 0 , ⳵ v / ⳵ x 0 , ⳵ x/ ⳵ v 0 , and ⳵ v / ⳵ v 0 . To begin, we take the derivative with respect to x 0 of both sides of Eq. 共25a兲,

⳵x 共 x , v ,t 兲 ⫽1⫹ ⳵x0 0 0

冕 ␶冕 t

␶

d

0

0

d␶⬘

⳵x 共 x , v , ␶ ⬘ 兲 T x共 x 共 x 0 , v 0 , ␶ ⬘ 兲 , ␶ ⬘ 兲 , ⳵x0 0 0

共37兲

where T x (x,t)⬅( ⳵ T/ ⳵ x)(x,t). Now, let us multiply both sides of Eq. 共37兲 by T x (x(x 0 , v 0 ,t),t), rename the variable t as t ⬘ and integrate in dt ⬘ from 0 to t. We obtain g 共 x 0 , v 0 ,t 兲 ⫽h 共 x 0 , v 0 ,t 兲 ⫹

冕

t

0

dt ⬘ T x 共 x 共 x 0 , v 0 ,t ⬘ 兲 ,t ⬘ 兲

冕

t⬘

0

d␶ g共 x0 ,v0 ,␶ 兲,

共38兲

where we have defined g 共 x 0 , v 0 ,t 兲 ⬅

冕

t

0

dt ⬘

⳵x 共 x , v ,t ⬘ 兲 T x 共 x 共 x 0 , v 0 ,t ⬘ 兲 ,t ⬘ 兲 ⳵x0 0 0

共39兲

and h 共 x 0 , v 0 ,t 兲 ⬅

冕

t

0

dt ⬘ T x 共 x 共 x 0 , v 0 ,t ⬘ 兲 ,t ⬘ 兲 .

共40兲

The second term on the right-hand side of Eq. 共38兲 is then integrated by parts, yielding

冕

t

0

dt ⬘ T x 共 x 共 x 0 , v 0 ,t ⬘ 兲 ,t ⬘ 兲

冕

t⬘

0

d ␶ g 共 x 0 , v 0 , ␶ 兲 ⫽h 共 x 0 , v 0 ,t 兲 ⫺ ⫽

冕

冕

t

0

t

0

冕

t

0

dt ⬘ g 共 x 0 , v 0 ,t ⬘ 兲

dt ⬘ g 共 x 0 , v 0 ,t ⬘ 兲 h 共 x 0 , v 0 ,t ⬘ 兲

dt ⬘ g 共 x 0 , v 0 ,t ⬘ 兲

冕

t

t⬘

d ␶ T x共 x 共 x 0 , v 0 , ␶ 兲 , ␶ 兲 . 共41兲

Thus, Eq. 共38兲 can be written as a Volterra equation, i.e., g 共 x 0 , v 0 ,t 兲 ⫽h 共 x 0 , v 0 ,t 兲 ⫹

冕

t

0

dt ⬘ K 共 x 0 , v 0 ,t,t ⬘ 兲 g 共 x 0 , v 0 ,t ⬘ 兲 ,

共42兲

where the ‘‘kernel’’ K(x 0 , v 0 ,t,t ⬘ ) is given by K 共 x 0 , v 0 ,t,t ⬘ 兲 ⬅

冕

t

t⬘

d ␶ T x共 x 共 x 0 , v 0 , ␶ 兲 , ␶ 兲 .

共43兲

Due to the integrability of 兩 T x 兩 ,

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J. Math. Phys., Vol. 40, No. 8, August 1999

兩 K 共 x 0 , v 0 ,t,t ⬘ 兲 兩 ⭐

冕

t

t⬘

Time-asymptotic traveling-wave solutions to . . .

d ␶ 兩 T x共 x 共 x 0 , v 0 , ␶ 兲 , ␶ 兲兩 ⭐

冕

⬁

t⬘

3903

d ␶ 兩 T x 共 x 共 x 0 , v 0 , ␶ 兲 , ␶ 兲 兩 ⬅H 共 x 0 , v 0 ,t ⬘ 兲 . 共44兲

Hence, Eq. 共42兲 implies the inequality 兩 g 共 x 0 , v 0 ,t 兲 兩 ⭐ 兩 h 共 x 0 , v 0 ,t 兲 兩 ⫹

冕

t

0

dt ⬘ H 共 x 0 , v 0 ,t ⬘ 兲 兩 g 共 x 0 , v 0 ,t ⬘ 兲 兩 .

共45兲

Then, from Gronwall’s inequality it follows that 兩 g 共 x 0 , v 0 ,t 兲 兩 ⭐C 共 x 0 , v 0 兲 e B 共 x 0 , v 0 兲 ⬅M 共 x 0 , v 0 兲 ⬍⬁,

共46兲

where C共 x0 ,v0兲⭐

冕

⬁

0

dt ⬘ 兩 T x 共 x 共 x 0 , v 0 ,t ⬘ 兲 ,t ⬘ 兲 兩

共47兲

冕

共48兲

and B共 x0 ,v0兲⬅

⬁

0

dt ⬘ H 共 x 0 , v 0 ,t ⬘ 兲 .

Now, let us go back to Eq. 共37兲, and to the corresponding equation for ⳵ v / ⳵ x 0 which is obtained by taking the partial derivative of Eq. 共25b兲 with respect to x 0 . According to Eq. 共39兲, these equations can be written in the form

⳵x 共 x , v ,t 兲 ⫽1⫹ ⳵x0 0 0

冕

t

0

d␶ g共 x0 ,v0 ,␶ 兲,

⳵v 共 x , v ,t 兲 ⫽g 共 x 0 , v 0 ,t 兲 . ⳵x0 0 0

共49a兲

共49b兲

From the bound on 兩 g(x 0 , v 0 ,t) 兩 in Eq. 共46兲 it follows immediately that

冏

冏

⳵x 共 x , v ,t 兲 ⭐1⫹M t, ⳵x0 0 0

冏

冏

⳵v 共 x , v ,t 兲 ⭐M . ⳵x0 0 0

共50a兲

共50b兲

Here, M is defined as supx 0 , v 0 M (x 0 , v 0 ) which exploits the fact that M (x 0 , v 0 ) is a uniformly bounded function of (x 0 , v 0 ), as follows from Eqs. 共46兲, 共47兲, 共48兲, and from the integrability properties of T x . Equations 共50兲 give the result sought: under the conditions on the time integrability of T x (x,t), we have found that ⳵ x/ ⳵ x 0 and ⳵ v / ⳵ x 0 can grow at most linearly in time. Using a completely analogous procedure to find similar bounds on the derivatives ⳵ x/ ⳵ v 0 and ⳵ v / ⳵ v 0 we obtain

冏 冏

冏 冏

⳵x 共 x , v ,t 兲 ⭐Nt, ⳵v0 0 0

共51a兲

⳵v 共 x , v ,t 兲 ⭐N. ⳵v0 0 0

共51b兲

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3904

J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

Here, N⬅supx 0 , v 0 N(x 0 , v 0 ), where N 共 x 0 , v 0 兲 ⬅1⫹D 共 x 0 , v 0 兲 e B 共 x 0 , v 0 兲 .

共52兲

B(x 0 , v 0 ) is the same quantity that was defined in Eq. 共48兲 and D(x 0 , v 0 ) satisfies D共 x0 ,v0兲⭐

冕

⬁

0

dt ⬘ t ⬘ 兩 T x 共 x 共 x 0 , v 0 ,t ⬘ 兲 ,t ⬘ 兲 兩 .

共53兲

From the inequalities in Eqs. 共50兲 and 共51兲, we easily obtain similar bounds on the time growth of the derivatives of the ‘‘inverse’’ functions x 0 and v 0 . These bounds follow from the fact that the Jacobian matrix of 关 x 0 (x, v ,t), v 0 (x, v ,t) 兴 共viewed as a function from R2 onto R2 兲 is the inverse of the Jacobian of the inverse function 关 x(x 0 , v 0 ,t), v (x 0 , v 0 ,t) 兴 for each given t, and that this latter has determinant equal to one. We find

冏 冏 冏 冏

冏 冏 冏

⳵x0 共 x, v ,t 兲 ⭐N, ⳵x

共54a兲

⳵v0 共 x, v ,t 兲 ⭐M , ⳵x

共54b兲

⳵x0 共 x, v ,t 兲 ⭐Nt, ⳵v

共55a兲

冏

共55b兲

⳵v0 共 x, v ,t 兲 ⭐1⫹M t. ⳵v

By combining the decay properties of T with Eqs. 共55兲, it is easy to see that the integrands T( ⳵ x 0 / ⳵ v ) and T( ⳵ v 0 / ⳵ v ) in the integrals in Eqs. 共35兲 and 共36兲 are integrable over R⫹ . Hence, by adding and subtracting the corresponding integrals on 关 t,⬁), we can write ˜ 共 x⫺ v t, v 兲 , x 0 共 x, v ,t 兲 ⬇x⫺ v t⫹G

共56a兲

˜ 共 x⫺ v t, v 兲 , v 0 共 x, v ,t 兲 ⬇ v ⫹H

共56b兲

where the symbol ⬇ is used to indicate that the two sides are equal up to certain transient functions of (x, v ,t) that disappear in the time-asymptotic limit, uniformly with respect to x and v . ˜ and H ˜ are given by The functions G ˜ 共 x, v 兲 ⬅ G

冕

˜ 共 x, v 兲 ⬅ H

冕

⬁

0 ⬁

0

d ␶ T 共 x⫹ v ␶ , ␶ 兲

⳵x0 共 x⫹ v ␶ , v , ␶ 兲 , ⳵v

共57a兲

d ␶ T 共 x⫹ v ␶ , ␶ 兲

⳵v0 共 x⫹ v ␶ , v , ␶ 兲 , ⳵v

共57b兲

and are clearly continuous 共x 0 and v 0 being C1 according to standard theorems on ordinary differential equations兲. 䊐 Equations 共32兲 tell us that the inverse particle trajectories at long times tend asymptotically to the straight-line paths (x⫺ v t, v ), but with one important qualification. Because of the effects of the transient, the phase point that is mapped backwards along the each straight-line trajectory is ˜ (x, v ), v T ⬅ v ⫹H ˜ (x, v ). Of course, the two not (x, v ) itself, but the ‘‘surrogate’’ point x T ⬅x⫹G ˜ and H ˜ are not known explicitly, since that would require the complete solution of the functions G nonlinear Vlasov equations for x T0 and v T0 , Eqs. 共33兲 and 共34兲. The essential point, here, is that

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

3905

these functions exist and do not depend explicitly on t. It is worth noting that there is a relationship ˜ ,H ˜ and the functions G,H introduced in Eqs. 共27兲 and 共30兲. Equations 共56兲 between the functions G imply that ˜ 共 x 共 x 0 , v 0 ,t 兲 ⫺ v共 x 0 , v 0 ,t 兲 t, v 兲 , x 0 ⬇x 共 x 0 , v 0 ,t 兲 ⫺ v共 x 0 , v 0 ,t 兲 t⫹G

共58a兲

˜ 共 x 共 x 0 , v 0 ,t 兲 ⫺ v共 x 0 , v 0 ,t 兲 t, v 兲 . v 0 ⬇ v共 x 0 , v 0 ,t 兲 ⫹H

共58b兲

Taking the limit t˜⬁ and using Eq. 共31兲 yields ˜ 共 x⬁ ,v⬁兲, x 0 ⫽x ⬁ ⫹G

共59a兲

˜ 共 x⬁ ,v⬁兲, v 0 ⫽ v ⬁ ⫹H

共59b兲

where (x ⬁ , v ⬁ ) are the quantities defined in Eqs. 共27兲 and 共30兲. Substituting Eqs. 共32兲 into the initial condition F␣ yields the solution f ␣T to Eq. 共24兲 in the form f ␣T 共 x, v ,t 兲 ⫽F␣T 共 x⫺ v t, v 兲 ⫹g ␣T 共 x, v ,t 兲 ,

共60兲

˜ 共 x, v 兲 , v ⫹H ˜ 共 x, v 兲兲 , F␣T 共 x, v 兲 ⬅F␣ 共 x⫹G

共61兲

g ␣T 共 x, v ,t 兲 ⫽F␣ 共 x T0 , v T0 兲 ⫺F␣T 共 x⫺ v t, v 兲 .

共62兲

where

and

Clearly, g ␣T (x, v ,t)˜0 uniformly as t˜⬁, since according to Eqs. 共32兲 the two terms on the right-hand side of Eq. 共62兲 become identical in this limit. This means that, as t goes to infinity, f ␣T can be obtained by advection along straight-line trajectories, as long as we replace the initial condition F␣ by the modified function F␣T that contains the effects of the transient field T. It follows from Eq. 共60兲 that f ␣T can be replaced in Eq. 共23兲 by a spatially uniform equilibrium that yields the same values for macroscopic quantities in the time-asymptotic limit. To be explicit, let us consider any integral of the form

冕

R

duG共 v ,u 兲 f ␣T 共 x,u,t 兲 ,

共63兲

which can be either a charge or current density (G( v ,u)⫽1,u), or any higher moment (G( v ,u) ⫽u n ), or a filtered distribution function.19 Substituting Eq. 共60兲 into Eq. 共63兲 and introducing the spatial Fourier series of F␣T , it is easy to see that for all k⫽0, lim t˜⬁

冕

R

duG共 v ,u 兲 f ␣T ,k 共 u,t 兲 ⫽ lim

t˜⬁

冕

R

duG共 v ,u 兲 F␣T ,k 共 u 兲 e ⫺ikut ⫽0

共64兲

冕

共65兲

by the Riemann–Lebesgue Lemma. Hence, lim t˜⬁

冕

R

duG共 v ,u 兲 f ␣T 共 x,u,t 兲 ⫽

R

duG共 v ,u 兲 F ␣T 共 u 兲 ,

where

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3906

J. Math. Phys., Vol. 40, No. 8, August 1999

F ␣T 共 v 兲 ⫽F␣T ,0共 v 兲 ⫽

1 2␲

冕

⫹␲

⫺␲

C. Lancellotti and J. J. Dorning

˜ 共 x, v 兲 , v ⫹H ˜ 共 x, v 兲兲 . dxF␣ 共 x⫹G

共66兲

The charge and current densities associated with F ␣T are

␳ T⫽

j T⫽

兺␣ q ␣ 冕Rd v 冕⫺ ␲ dxF␣共 x⫹G˜ 共 x, v 兲 , v ⫹H˜ 共 x, v 兲兲 ,

共67兲

兺␣ q ␣ 冕Rd v 冕⫺ ␲ dx v F␣共 x⫹G˜ 共 x, v 兲 , v ⫹H˜ 共 x, v 兲兲 .

共68兲

1 2␲

1 2␲

⫹␲

⫹␲

˜ (x, v ), v ⫹H ˜ (x, v ) 兴 and noting that this Changing the integration variables to (x 0 , v 0 )⫽ 关 x⫹G transformation preserves the area in phase space 共since it is the time-limit of a Hamiltonian flow兲, we get

兺␣ q ␣ 冕Rd v 0 冕⫺ ␲ dx 0 F␣共 x 0 , v 0 兲 ,

共69兲

兺␣ q ␣ 冕Rd v 0 冕⫺ ␲ dx 0 关v 0 ⫹H 共 x 0 , v 0 兲兴 F␣共 x 0 , v 0 兲 .

共70兲

␳ T⫽

j T⫽

1 2␲

1 2␲

⫹␲

⫹␲

Since the spatially uniform part of the initial condition F␣ (x, v ) has been taken to be a Vlasov equilibrium, it is easy to see that Eqs. 共69兲 and 共70兲 reduce to

␳ T ⫽0,

j T ⫽ j ⬁T ,

共71兲

兺␣ q ␣ 冕Rd v 0 冕⫺ ␲ dx 0 H 共 x 0 , v 0 兲 F␣共 x 0 , v 0 兲 .

共72兲

where j ⬁T ⬅

1 2␲

⫹␲

Whenever T is a solution of the self-consistent VPA equation, Eq. 共12兲, j ⬁T ⫽0, as follows directly from the time-and-space averaged Ampe`re equation, Eq. 共21兲. In general, let us define the subspace TL 傺T of the transient fields that satisfy the hypotheses of Theorem 1 and for which j ⬁T ⫽0. Then, Eq. 共71兲 leads directly to the main result of this section: Theorem 2: For any choice of F␣ and T苸TL the time-asymptotic equation, Eq. (22a), has the solution A⬅0. V. VANISHING ASYMPTOTIC STATES

Theorem 2 can be given the following interpretation: the asymptotic equation, Eq. 共22a兲, possesses an infinite dimensional manifold of vanishing solutions with respect to the initial condition F␣ (x, v ), viewed as an infinite dimensional parameter in the Banach space S of all the C1 (R⫻R) functions such that 储 F␣ 储 S⬅sup x

冕

R

d v共 1⫹ 兩 v 兩 兲 兩 F␣ 共 x, v 兲 兩 ⬍⬁.

共73兲

In Eq. 共6兲 F␣ (x, v ) was written as the sum of a spatially uniform Vlasov equilibrium F ␣ ( v ) and another function h ␣ (x, v ); in many situations it is convenient to consider F ␣ 苸S as fixed and study how the solutions to the VPA problem depend on h ␣ . Physically, this corresponds to assuming a given ‘‘background equilibrium’’ and varying a spatially dependent perturbation. Theorem 2 tells us that A⬅0 is a solution to Eq. 共22a兲 for all h ␣ 苸S. Schematically, if we consider the h ␣ -A plane

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

3907

共where each axis represents an infinite dimensional space兲 this ‘‘basic solution branch’’ A⬅0 can be drawn as the horizontal line along the h ␣ axis. We shall call these solutions vanishing asymptotic states. It is important to emphasize that these vanishing asymptotic states do not necessarily correspond to solutions of the complete system, Eqs. 共22兲. In fact, it is not at all certain that if A⫽0 is substituted into Eq. 共22b兲 the equation that results for T from Eq. 共23兲, T⫽N共 T,F ␣ ,h ␣ 兲

共74兲

will possess a solution in T. Whenever it does possess such a solution for an initial perturbation h ␣ , we shall say that the corresponding point on the zero solution branch for the asymptotic equation, Eq. 共22a兲, is accessible to the system. Clearly, at least one vanishing asymptotic state is always accessible, namely the ‘‘origin’’ A⬅h ␣ ⬅0. Any other accessible vanishing asymptotic state 共a.v.a.s. from now on兲 will correspond, physically, to strongly Landau damped evolution,6 with the field damping completely to zero before trapping effects are able to sustain travelingwave propagation. Let us consider a generic a.v.a.s. A⬅0, h ␣ ⫽h ␣0 , associated with a transient field T⫽T 0 such that the nonlinear Ampe`re equation, Eq. 共74兲 is satisfied. The corresponding distribution function T T f ␣0 will be the solution of the Vlasov equation, Eq. 共24兲, with the initial condition f ␣0 (x, v ,0) ⬅F ␣0 (x, v )⬅F ␣ ( v )⫹h ␣0 (x, v ), where F ␣ ( v ) is a given Vlasov equilibrium. We now introduce the following definition: Definition 1: An a.v.a.s. (A,T,h ␣ )⫽(0,T 0 ,h ␣0 ) will be called critical if in every 共arbitrarily small兲 neighborhood of (0,T 0 ,h ␣0 ) in AP⫻T⫻S there is a point (A,T 0 ⫹ ␦ T,h ␣0 ⫹ ␦ h ␣ ) such that the initial perturbation h ␣0 ⫹ ␦ h ␣ generates a solution to the VPA problem in which the electric field has transient part T 0 ⫹ ␦ T and nonzero time-asymptotic part A. Obviously, this situation is physically very interesting, especially in the class of problems that fall under the label of ‘‘nonlinear Landau damping;’’ 20 in these cases, we expect an a.v.a.s. of this kind to mark the transition between solutions that Landau damp completely and solutions that contain a nonzero small-amplitude asymptotic part. Another important example of a critical a.v.a.s. is found whenever the background equilibrium F ␣ allows undamped traveling wave solutions for perturbations of arbitrarily small amplitude. Examples are the recently discovered smallamplitude BGK and BGK-type solutions.10,12 In these cases, of course, the critical a.v.a.s. is given by the ‘‘origin’’ (A,T,h ␣ )⫽(0,0,0). Now, the fact that A⬅0 provides a solution to the asymptotic equation in isolation for any choice of h ␣ , suggests that the asymptotic equation itself may be amenable to a bifurcation analysis at the critical states. In fact, if we consider the asymptotic equation alone, in an arbitrarily small neighborhood of a critical a.v.a.s. the solution to the equation cannot be unique, since from Definition 1 both A⬅0 and the nonzero asymptotic solutions to the VPA system corresponding to h ␣0 ⫹ ␦ h ␣ satisfy the equation. Thus, according to a rather general definition in nonlinear analysis 共e.g., Ref. 21, p. 151兲, A⬅0, h ␣ ⫽h ␣0 is a bifurcation point for the asymptotic equation. Of course, this is not true in general for the complete VPA system, for example, it is not true whenever the solution to the initial value problem is known to be unique. In summary, we have the following theorem: Theorem 3: Every critical a.v.a.s for the VPA initial value problem corresponds to a bifurcation point for the asymptotic Ampe`re equation, Eq. (22a). VI. TIME-ASYMPTOTIC LINEAR ANALYSIS

Let us now consider a given critical a.v.a.s. ␩ 0 ⬅(0,T 0 ,h ␣ ). Instead of carrying out the bifurcation analysis of Eq. 共22a兲 at ␩ 0 directly, we shall study the equivalent problem comprised of the countable set of equations for the Fourier–Bohr coefficients of A, a k, ␻ i ⫽Nk, ␻ i 共 A⫹T,F ␣ ,h ␣ 兲 .

共75兲

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3908

J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

These equations correspond to all the k and ␻ i such that a k, ␻ i ⫽0; by definition, the Fourier–Bohr coefficients Nk, ␻ i of the nonlinear operator N coincide with those of P a N. The double-indexed sequences y⬅ 兵 y m, ␩ 其 of Fourier–Bohr coefficients of functions that are periodic in space and a.p.c. in time are characterized by the Riesz-type condition22 ⌺ m, ␩ 兩 y m, ␩ 兩 2 ⬍⬁, where it is understood that the index-set for ␩ can be any countable subset of the real axis R, and not just the integers. With the supnorm, 储 y储 ⫽sup兩 y m, ␩ 兩 ,

共76兲

m, ␩

the set of all the y is a Banach space, which we shall denote l b . The sequence of nonlinear time-asymptotic equations, Eq. 共75兲, can be reformulated as a single functional equation in l b by writing a⫽ 兵 a k, ␻ i 其 for the double sequence of the Fourier–Bohr coefficients of A and defining N(a)⬅ 兵 Nk, ␻ i (A⫹T,F ␣ ,h ␣ ) 其 . Then, Eq. 共75兲 becomes 共77兲

a⫽N 共 a兲 .

Here, we are not explicitly writing the dependence of N on the transient field T, which is being treated as a free parameter along with F ␣ and h ␣ . In a local analysis, the natural question is whether there is any appropriate linear approximation to the nonlinear problem under consideration. We answer this question by first establishing the following result: Lemma 1: For E苸W, F␣ 苸S and k⫽0, Nk, ␻ i 共 E,F ␣ ,h ␣ 兲 ⫽

2 k

冕 dt 冕⫺␲ dx e ⫺i␻ t⫺ikxE共 x,t 兲 P冕Rd v 兺␣ m ␣␣ ␴lim ˜⬁ ␴ 0

⫽

2 k

lim lim dt 冕 dx e ⫺i ␻ t⫺ikx E 共 x,t 兲 冕 冕 兺␣ m ␣ n˜⬁ ␴ 0 ⫺␲ ⍀ ␴ ˜⬁

q2

␴

1

⫹␲

i

q ␣2

1

␴

f␣⬘ 共 E,F␣ 兲

␻ i ⫹k v

⫹␲

i

c n,i

dv

f␣⬘ 共 E,F␣ 兲

␻ i ⫹k v

, 共78兲

c represents the real axis R where f␣⬘ (E,F␣ )⫽ ⳵ f ␣ / ⳵ v (x, v ,t) was defined in Eq. (11) and ⍀ n,i minus the one-dimensional sphere of radius r n ⬍1/n centered at ⫺ ␻ i /k, ⍀ n,i ⬅B 关 ⫺( ␻ i /k),r n 兴 . Proof: The Fourier–Bohr coefficients Nk, ␻ i are given by

Nk, ␻ i 共 E,F ␣ ,h ␣ 兲 ⫽

4␲ ik

1

兺␣ q ␣ ␴lim ˜⬁ ␴

冕

␴

dt e ⫺i ␻ i t

0

冕

R

d v f ␣ ,k 共 v ,t 兲 ,

共79兲

The properties of f ␣ enable us to apply Fubini’s theorem and rewrite this equation as Nk, ␻ i 共 E,F ␣ ,h ␣ 兲 ⫽

4␲ ik

lim 冕 兺␣ q ␣ ␴lim ˜⬁ n˜⬁ ⍀

c n,i

dv

1 ␴

冕

␴

0

dt e ⫺i ␻ i t f ␣ ,k 共 v ,t 兲 .

共80兲

We substitute into the right-hand side of Eq. 共80兲 the expression for f ␣ that is given by the Vlasov equation in integro-differential Fourier-transformed form f ␣ ,k 共 v ,t 兲 ⫽h ␣ ,k 共 v 兲 e ⫺ik v t ⫺

q␣ m␣

冕

t

0

d␶

1 2␲

冕

⫹␲

⫺␲

dxe ⫺ik 关 x⫹ v共 t⫺ ␶ 兲兴 E 共 x, ␶ 兲

⳵f␣ 共 x, v , ␶ 兲 . ⳵v

共81兲

The term proportional to h ␣ ,k ( v ) does not contribute to Eq. 共80兲 by the Riemann–Lebesgue Lemma. After an integration by parts, we are left with

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J. Math. Phys., Vol. 40, No. 8, August 1999

⫺

2 k

2 ⫹ k

q ␣2

兺␣ m ␣ ␴ ˜⬁ q ␣2

lim

1 P ␴

冕 ␻ 冕 冕 冕 ␻ 冕

1 lim P m ␣ ␴ ˜⬁ ␴

兺␣

Time-asymptotic traveling-wave solutions to . . .

R

e ⫺i ␻ i ␴ i ⫹k v

dv

␴

R

dv

0

␴

dte ⫺ik v共 ␴ ⫺t 兲

0

e ⫺i ␻ i t dt i ⫹k v

⫹␲

⫺␲

冕

⫹␲

⫺␲

dxe ⫺ikx E 共 x,t 兲

dxe ⫺ikx E 共 x,t 兲

3909

⳵f␣ 共 x, v ,t 兲 ⳵v

⳵f␣ 共 x, v ,t 兲 . ⳵v

共82兲

Here, the two v -integrals, taken separately, would not be well-defined if the principal values had not been introduced via Eq. 共80兲. The first term in Eq. 共82兲 can be greatly simplified by noting that it contains the last term on the right-hand side of Eq. 共81兲 evaluated at t⫽ ␴ . Hence, that whole term vanishes in the limit ␴ ˜⬁, since 1/␴ multiplies functions that are bounded in ␴. Finally, via another change in the order of integration 共based on Fubini’s theorem, and also on the Lebesgue dominated convergence theorem兲 in order to bring the limit n˜⬁ inside the integrals in dt and dx, we are left with the expression for the Nk, ␻ i on the first line in Eq. 共78兲. In order to obtain the second expression in Eq. 共78兲, we need to change the order of the limits ␴ ˜⬁ and n˜⬁. This can be done most conveniently in Eq. 共79兲 in order to take advantage of c , the properties of f ␣ . Thus, we break the v integration in Eq. 共79兲 into two parts, on ⍀ n,i and ⍀ n,i respectively. As n˜⬁, the part on ⍀ n,i vanishes, since

冏 冕 1 ␴ ˜⬁ ␴

␴

lim

dte ⫺i ␻ i t

0

冕

⍀ n,i

冏

d v f ␣ ,k 共 v ,t 兲 ⭐

2M ␣ n

共83兲

due to the a priori bound in Eq. 共2兲. Taking the limit n˜⬁ we obtain the following expression for Nk, ␻ i (E,F ␣ ,h ␣ ): Nk, ␻ i 共 E,F ␣ ,h ␣ 兲 ⫽

4␲ ik

lim lim 冕 兺␣ q ␣ n˜⬁ ␴ ˜⬁ ⍀

c n,i

dv

1 ␴

冕

␴

0

共84兲

dt f ␣ ,k 共 v ,t 兲 .

Then, the same procedure that was applied to Eq. 共80兲 共with minor adaptations兲 leads to the expression on the second line in Eq. 共78兲. 䊐 The fact that the order of the limits n˜⬁ and ␴ ˜⬁ in Eq. 共78兲 can be changed is crucial for the next Lemma, which shows that the explicit dependence on E in Eq. 共78兲 actually involves only the asymptotic part A, while the term containing T vanishes. 共Of course, the transient T still appears implicitly through the distribution function f ␣ , which is determined by both A and T via the Vlasov equation.兲 Lemma 2: For any choice of E苸W, F␣ 苸S and k⫽0, 2 k

lim lim 冕 dt e ⫺i␻ t 冕⫺␲ dx e ⫺ikxT共 x,t 兲 冕⍀ 兺␣ m ␣ n˜⬁ ␴ ˜⬁ ␴ 0 q ␣2

1

␴

⫹␲

i

c n,i

dv

f␣⬘ 共 E,F␣ 兲

␻ i ⫹k v

⫽0.

共85兲

Proof: Integrating by parts gives

冏冕

冏冏冋

⳵f␣ 共 x, v ,t 兲 ⳵v f ␣ 共 x, v ,t 兲 d ⫽ ⫺ v c ␻ i ⫹k v ␻ i ⫹k v ⍀ n,i ⭐

4␲M ␣ k ⫹ 2 rn rn

册

冕

⫺ 共 ␻ i /k 兲 ⫹r n ⫺ 共 ␻ i /k 兲 ⫺r n

R

d v 兩 f ␣兩 ⭐

⫹k

冕

c ⍀ n,i

dv

冏

f ␣ 共 x, v ,t 兲 共 ␻ i ⫹k v 兲 2

˜ 4 ␲ M ␣ kM ⫹ 2 ⬅M ␣ ,n , rn rn

共86兲

where we have exploited the a priori bound on f ␣ (x, v ,t) and the fact that the total number of ˜ must be conserved 共as follows from integrating the Vlasov equation over the whole particles M phase plane兲. Hence, in the second line of Eq. 共78兲,

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3910

J. Math. Phys., Vol. 40, No. 8, August 1999

冏

1 ␴ ˜⬁ ␴ lim

冕

␴

dt e ⫺i ␻ i t

0

1 ␴ ˜⬁ ␴

⭐ lim

冕

␴

0

冕

⫹␲

⫺␲

C. Lancellotti and J. J. Dorning

dx e ⫺ikx T 共 x,t 兲

冕

c ⍀ n,i

dv

冏

f␣⬘ 共 E,F␣ 兲

␻ i ⫹k v

dt 兩 T 共 x,t 兲 兩 2 ␲ M ␣ ,n ⫽2 ␲ M ␣ ,n Mt 关 兩 T 共 x,t 兲 兩 兴 ⫽0,

共87兲

where the mean value Mt was defined in Eq. 共17兲 and Mt 关 兩 T(x,t) 兩 兴 is zero since T(x,t) is a transient. Thus Lemma 2 is proved. Clearly since M ␣ ,n diverges in the limit n˜⬁, the order in which the two limits are taken in Eq. 共85兲 is crucial. 䊐 According to Eqs. 共78兲 and 共85兲, when k⫽0 the Fourier–Bohr coefficients Nk, ␻ i can be rewritten as

冕

1 ␴ ˜⬁ ␴

Nk, ␻ i 共 A⫹T,F ␣ ,h ␣ 兲 ⫽ lim

␴

冕

1 2␲

dt

0

⫹␲

dxe ⫺i ␻ i t⫺ikx AE共 k, ␻ i 兲 共 A⫹T,F ␣ ,h ␣ 兲 ,

⫺␲

共88兲

where we have introduced the notation 共 k, ␻ i 兲

E

4␲ 共 E,F ␣ ,h ␣ 兲 ⬅ k

兺␣ m ␣ P 冕Rd v q ␣2

f␣⬘ 共 E,F␣ 兲

␻ i ⫹k v

.

共89兲

We next consider k⫽0 which we excluded from Lemmas 1 and 2. There are two possibilities, ␻ i ⫽0 and ␻ i ⫽0. However, it is not necessary to study the former case since a 0,0⫽0 follows immediately from the fact that the VPA system conserves energy. Indeed, if a 0,0 were nonzero the corresponding time-asymptotic electric field, being uniform in both space and time, clearly would accelerate all the particles to infinite energies. Thus, we do not consider this case; for the same reason, we exclude the possibility of ␻ ⫽0 being an accumulation point for the ␻ i . Then, for k ⫽0 there must exist a ␭苸R such that k⫽0, ␻ i ⭐␭⇒a k, ␻ i ⫽0, which in fact is a well-known sufficient condition for the spatial average A 0 (t) to have an almost periodic primitive 共Ref. 17, p. 74兲. Hence, we next consider the remaining case, k⫽0, ␻ i ⫽0, under this condition. The analogues of Lemmas 1 and 2 are developed immediately using the same arguments employed in these Lemmas, with Eq. 共81兲 for the kth Fourier component replaced by f ␣ ,0共 v ,t 兲 ⫽F ␣ 共 v 兲 ⫺

q␣ m␣

冕

t

0

d␶

1 2␲

冕

⫹␲

⫺␲

dx E 共 x, ␶ 兲

⳵f␣ 共 x, v , ␶ 兲 , ⳵v

共90兲

which is obtained via the time integration of the k⫽0 component of the Vlasov equation. Substituting Eq. 共90兲 into Eq. 共4兲 and carrying out steps analogous to those implemented above for k ⫽0 leads to an expression identical to Eq. 共88兲, with E共 0,␻ i 兲 共 E,F ␣ ,h ␣ 兲 ⬅

4␲

␻ 2i

兺␣ m ␣ 冕Rd v f␣共 E,F␣ 兲 . q ␣2

共91兲

Thus, Lemmas 1 and 2 have been extended to include the case k⫽0. Equation 共88兲 combined with Eq. 共89兲 for k⫽0 and Eq. 共91兲 for k⫽0 leads directly to the following theorem; Theorem 4: Given a critical a.v.a.s. ␩ 0 苸AP⫻T⫻S, let E(k, ␻ i ) (E,F ␣ ,h ␣ ) be continuous in E at ␩ 0 . Then, Nk, ␻ i (A⫹T,F ␣ ,h ␣ ) is Fre´chet-differentiable with respect to A at ␩ 0 , the derivative being 1 ␴ ˜⬁ ␴

Lk, ␻ i 共 T 0 ,F ␣ ,h ␣0 兲 A⬅ lim

冕

␴

0

dt

1 2␲

冕

⫹␲

⫺␲

dxe ⫺i ␻ i t⫺ikx AE共 k, ␻ i 兲 共 T 0 ,F ␣ ,h ␣0 兲 .

共92兲

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

3911

Proof: Clearly Lk, ␻ i is a continuous linear operator, and due to Eq. 共88兲 关Lemmas 1 and 2兴 and the continuity of E(k, ␻ i ) at ␩ 0 , 兩 Nk, ␻ i 共 T 0 ⫹A,F ␣ ,h ␣0 兲 ⫺Nk, ␻ i 共 T 0 ,F ␣ ,h ␣0 兲 ⫺Lk, ␻ i 共 T 0 ,F ␣ ,h ␣0 兲 A 兩

冏

1 ␴ ␴ ˜⬁

⫽ lim

冕

␴

0

dt

冕

1 2␲

⫹␲

⫺␲

dxe ⫺i ␻ i t⫺ikx A 关 E共 k, ␻ i 兲 共 T 0 ⫹A,F ␣ ,h ␣0 兲 ⫺E共 k, ␻ i 兲 共 T 0 ,F ␣ ,h ␣0 兲兴

⭐ 储 A 储储 E共 k, ␻ i 兲 共 T 0 ⫹A,F ␣ ,h ␣0 兲 ⫺E共 k, ␻ i 兲 共 T 0 ,F ␣ ,h ␣0 兲储 ⫽o 共 储 A 储 兲 .

冏 共93兲 䊐

The sequence Lk, ␻ i A defines a linear operator La⬅ 兵 Lk, ␻ i (T 0 ,F ␣ ,h ␣0 )A 其 on l b . Introducing M (a)⬅N(a)⫺La enables us to write Eq. 共77兲 in the form 共94兲

共 I⫺L 兲 a⫽M 共 a兲 .

The operator (I⫺L) represents the linear approximation 共at the given critical a.v.a.s.兲 to the original nonlinear equation, Eq. 共77兲. Hence, the first step in the analysis of the nonlinear problem will be the study of the invertibility of this linear operator. Specifically, we must determine the null space of (I⫺L); hence, we must study the linearized time-asymptotic equation 共95兲

共 I⫺L 兲 a⫽0.

Before proceeding with the general analysis of this equation, we consider a particularly simple case for which the analysis is straightforward. It is provided by the a.v.a.s. h ␣ ⬅0, i.e., the zero field solution T

A 共 x,t 兲 ⫽T 0 共 x,t 兲 ⬅0,

f ␣0 共 x, v ,t 兲 ⫽f␣ 共 0,F ␣ 兲 ⫽F ␣ 共 v 兲

共96兲

for which E(k, ␻ i ) (T 0 ,F ␣ ,h ␣0 ), which appears in Eq. 共92兲, can be calculated immediately. From Eq. 共89兲 and Eq. 共91兲 it is 共 k, ␻ i 兲

E

兺␣ m ␣ P冕Rd v ␻ i␣⫹k v

共97兲

兺␣ m ␣ 冕Rd v F ␣共 v 兲 ⫽ ␻ 2i

共98兲

4␲ 共 0,F ␣ ,0兲 ⬅ k

q ␣2

F ⬘共 v 兲

for k⫽0, and 共 0,␻ i 兲

E

共 0,F ␣ ,0兲 ⬅

4␲

␻ 2i

q ␣2

␻ 2p

for k⫽0, where ␻ p is the plasma frequency that corresponds to the equilibrium F ␣ ( v ). The expressions given in Eqs. 共97兲 and 共98兲 are both constants and can be moved out of the integrals in Eq. 共92兲. Thus, Eq. 共95兲 takes the simple form a k, ␻ i D0 共 k, ␻ i 兲 ⫽0,

共99兲

where D0 共 k, ␻ i 兲 ⬅1⫺

4␲ k

兺␣ m ␣ P冕Rd v ␻ i␣⫹k v q ␣2

F⬘ 共v兲

共100兲

for k⫽0 and D0 (0,␻ i )⬅ ␻ 2p / ␻ 2i 共for k⫽0兲. D0 (k, ␻ i ) is the well-known Vlasov dielectric function,23 and Eq. 共99兲 is trivial to solve; it implies that a k, ␻ i ⫽0 except for all the choices of k and ␻ i that satisfy the Vlasov dispersion relation

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3912

J. Math. Phys., Vol. 40, No. 8, August 1999

C. Lancellotti and J. J. Dorning

D0 共 k, ␻ i 兲 ⫽0.

共101兲

In general the linear operator in Eq. 共95兲 is determined by E(k, ␻ i ) (T 0 ,F ␣ ,h ␣0 ) through Eq. 共92兲, and E(k, ␻ i ) (T 0 ,F ␣ ,h ␣0 ) in turn is determined solely by the initial condition and the transient T T field T 0 via the distribution function f ␣0 at the a.v.a.s. However, f ␣0 generally will not be a simple Vlasov equilibrium because the transient T 0 is nonzero and affects the distribution function. Unfortunately, it is very difficult to solve exactly both the nonlinear Ampe`re equation for T 0 , Eq. T 共74兲, and the Vlasov equation for f ␣0 , Eq. 共24兲. However, it is possible, without doing this, to obtain important results on the asymptotic linear operator by exploiting the general properties of T particle motion in a transient field. If we define a Vlasov equilibrium F ␣0 according to Eq. 共66兲, T Eq. 共65兲 shows that this F ␣0 yields the same values for macroscopic quantities as the distribution T0 functions f ␣ in the time asymptotic limit. We now raise the question of whether these Vlasov T T equilibria F ␣0 yield the same values as f ␣0 for the integrals in Eqs. 共89兲 and 共91兲 and therefore the same linearized operator Lk, ␻ i . In fact, we have the following result: Theorem 5: At a given critical a.v.a.s. ␩ 0 , Lk, ␻ i 共 T 0 ,F ␣ ,h ␣ 兲 A⫽ 关 1⫺D共 k, ␻ i 兲兴 a k, ␻ i ,

共102兲

兺␣ m ␣ P冕Rd v ␻ i ⫹k v

共103兲

where 4␲ D共 k, ␻ i 兲 ⬅1⫺ k

T

F ␣0 ⬘ 共 v 兲

q ␣2

T

for k⫽0 and D(0,␻ i )⬅1⫺ ␻ 2p (T 0 )/ ␻ 2i (for k⫽0). Here F ␣0 ⬘ is the velocity derivative of the time-asymptotic Vlasov equilibrium defined in Eq. (66) and ␻ 2p (T 0 ) is the corresponding plasma frequency. Proof: We shall consider the case k⫽0; the case k⫽0 is straightforward. After substituting Eq. 共60兲 into Eq. 共92兲 and explicitly writing out the expression for E(k, ␻ i ) , Eq. 共89兲, we can change the order of the limits ␴ ˜⬁ and n˜⬁ according to Lemma 1. Then the term containing the T functions g ␣0 vanishes by Fre´chet’s Lemma16 and we are left with

4␲ k

lim lim 冕 dt 冕 兺␣ m ␣ n˜⬁ 2 ␲ ⫺␲ ␴ ˜⬁ ␴ 0 q ␣2

1

␴

1

⫹␲

⳵ T 关 F␣0 共 x⫺ v t, v 兲兴 ⳵ v dx e ⫺i ␻ i t⫺ikx A 共 x,t 兲 c d v . ␻ i ⫹k v ⍀ n,i

冕

共104兲

By applying the convolution theorem to the integral in dx this becomes

4␲ k

兺␣

q ␣2

1 lim lim m ␣ n˜⬁ ␴ ˜⬁ ␴

冕

␴

0

dte ⫺i ␻ i t

⳵ T 关 F 0 共 v 兲 e ⫺ik ⬘ v t 兴 ⳵ v ␣ ,k ⬘ A k⫺k ⬘ 共 t 兲 c d v . ␻ i ⫹k v ⍀ n,i k⬘

兺

冕

共105兲

It can be easily seen that the term corresponding to k ⬘ ⫽0 gives Eq. 共102兲. Hence, it will be enough to show that all the terms with k ⬘ ⫽0 are equal to zero in order to establish the result. An integration by parts gives

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J. Math. Phys., Vol. 40, No. 8, August 1999

⳵ T 关 F ␣0,k 共 v 兲 e ⫺ik ⬘ v t 兴 ⬘ ⳵v d ⫽k v c ␻ i ⫹k v ⍀ n,i

冕

冕

Time-asymptotic traveling-wave solutions to . . .

F ␣0,k 共 v 兲 e ⫺ik ⬘ v t T

c

⍀ n,i

dv

T

冉

⬘

共 ␻ i ⫹k v 兲

⫹F ␣0,k ⫺ ⬘

2

⫺

冊

冋 冉 册

3913

冊

␻i 1 T F␣0,k ⫺ ⫺r n e i 共共 k ⬘ /k 兲 ␻ i ⫹k ⬘ r n 兲 t ⬘ rn k

␻i ⫹r n e i 共共 k ⬘ /k 兲 ␻ i ⫺k ⬘ r n 兲 t . k

共106兲

Now, the first term on the right-hand side vanishes in the time-asymptotic limit by the Riemann– Lebesgue Lemma. When the other terms are substituted into Eq. 共105兲, they generate quantities ˜ i ⬅⫺(1⫺(k ⬘ /k)) ␻ i ⫾k ⬘ r n . Since proportional to the Fourier–Bohr coefficients a k⫺k ⬘ , ␻˜ i , where ␻ A is almost periodic, all these coefficients will be equal to zero except those corresponding to some very special choices of the radii r n such that ␻ ˜ i belongs to the countable set of nonzero frequencies of A. However, since the set of these ‘‘bad radii’’ has zero measure in R, we can always pick the sequence r n 共when introducing the family of spheres in Eq. 共78兲兲 in such a way that a k⫺k ⬘ , ␻˜ i ⫽0᭙n; thus, the limit for n˜⬁ in Eq. 共105兲 also will be zero, which proves the assertion. 䊐 The conclusion of this analysis is that the operator (I⫺L) can be written component-wise as 共 I⫺L 兲 a⬅ 兵 a k, ␻ i D共 k, ␻ i 兲 其 .

共107兲

T

Exploiting the regularity and integrability of F ␣0 , it is easy to see that the function D(k, ␻ i ) is ˆ such that bounded, i.e., there is a constant M ˆ. 兩 D共 k, ␻ i 兲 兩 ⭐M

共108兲

This implies that I⫺L maps l b into l b and is continuous, since

兺 兩 a k, ␻ D共 k, ␻ i 兲 兩 2 ⭐Mˆ 2 k,兺␻ 兩 a k, ␻ 兩 2 ⬍⬁

共109兲

ˆ 储 a储 . 储共 I⫺L 兲 a储 ⫽sup兩 a k, ␻ i D共 k, ␻ i 兲 兩 ⭐M

共110兲

k, ␻ i

i

i

i

and

k, ␻ i

The null space of I⫺L consists of those sequences whose only nonzero entries correspond to pairs of real indexes (k, ␻ i ) that satisfy the time-asymptotic Vlasov dispersion relation D共 k, ␻ i 兲 ⫽0

共111兲 T

which is determined by the asymptotic Vlasov equilibria F ␣0 corresponding to the transient field T 0 at the critical a.v.a.s. under consideration. It is important to note that in most physically relevant situations there is only a finite number N of such pairs. In these cases the linear operator I⫺L has a finite-dimensional null space K ⬅Ker(I⫺L), with dim(K)⫽N, and I⫺L is a Fredholm operator. A good example is given by the Vlasov dispersion curve for the Maxwellian 共Fig. 1兲, whose features are quite representative of what one encounters in most applications. There is a cutoff wave number k d such that for k⬎k d Eq. 共111兲 has no solution at all; hence, given any wave number k⭐k d , Eq. 共111兲 has a finite number N k of simple real roots ␭ 1 (k), ␭ 2 (k)¯␭ N k (k). Since the basic wave number is k⫽1, % N k possible there will be %⫽ 关 k d 兴 admissible wave numbers before cutoff, for a total of N⫽ 兺 k⫽1 modes.

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C. Lancellotti and J. J. Dorning

FIG. 1. The Vlasov dispersion curve for a Maxwellian e – p plasma with T e ⫽T p , and the roots ␻ (k) of the corresponding dispersion relation for a basic wave number k 0 and its harmonics; k is in units of the inverse Debye length k D ⬅1/␭ D and ␻ is in units of the plasma frequency ␻ p .

VII. TIME-ASYMPTOTIC FIELD SOLUTION

Now that we have carried out the analysis of the linearized time-asymptotic problem, we are ready for our final step, which is simply to show that the solution to the linearized problem provides 共as should be expected兲 a leading order solution for A to the nonlinear time-asymptotic equation, Eq. 共22a兲. Since the Fre´chet derivative at the a.v.a.s. has a nontrivial null-space and is not invertible, the simplest thing to do is to apply the classical method of analysis known as the Alternative Method.24 To do this, we decompose the space l b into the direct sum of K and its complement H. From Eq. 共107兲 it follows immediately that Rg(I⫺L)⫽ 关 Ker(I⫺L) 兴⬜ ⫽H. Let us consider the two projectors Q K and Q H associated, respectively, with K and H. Q K is the operator that starts from any element of l b and annihilates all the entries except for those that have one of the N pairs of indexes that satisfy the Vlasov dispersion relation. Conversely, Q H cancels only the entries that have such indexes. Let us define ⌿⬅Q Ka and ⌽⬅Q Ha, so that a⫽⌿⫹⌽. Then, a standard procedure24 leads from Eq. 共94兲 to the two equations, Q KM 共 ⌿⫹⌽ 兲 ⫽0,

共112兲

共 I⫺L 兲 ⌽⫽Q HM 共 ⌿⫹⌽ 兲 .

共113兲

In the language of bifurcation theory, Eq. 共112兲 is the bifurcation equation, while Eq. 共113兲 is the auxiliary equation. It is easy to verify that when the operator I⫺L is restricted to the subspace H, it is invertible; from Eq. 共107兲 it follows immediately that given b苸H, 共 I⫺L 兲 ⫺1 b⫽

再

b k, ␻ i D共 k, ␻ i 兲

冎

,

共114兲

where D(k, ␻ i )⫽0 ᭙b k, ␻ i since b苸H. Then, the auxiliary equation, Eq. 共113兲 can be written as a fixed point problem for ⌽, ⌽⫽ 共 I⫺L 兲 ⫺1 Q HM 共 ⌿⫹⌽ 兲 .

共115兲

Here, we are not interested in a detailed fixed point analysis of this equation, since we have been assuming from the beginning that an asymptotically almost periodic solution to the VPA problem

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J. Math. Phys., Vol. 40, No. 8, August 1999

Time-asymptotic traveling-wave solutions to . . .

3915

does exist. In fact, the existence and uniqueness of the solution for A implies that the solutions for ⌽ and ⌿ must also exist and be unique. What is important is that Eqs. 共112兲 and 共115兲 enable us to write the general form of the nonlinear solution for the time-asymptotic field A, as expressed in the following theorem: Theorem 6: Let ␩ 苸AP⫻T⫻S be a critical a.v.a.s., and let E(k, ␻ i ) (E,F ␣ ,h ␣ ) be continuous in E at ␩. Then, the general solution to the time-asymptotic equation, Eq. (22a), in a neighborhood of ␩ in AP⫻T⫻S is given by A 共 x,t 兲 ⫽

兺

k, ␻ i

␺ k, ␻ i e ikx⫹i ␻ i t ⫹o 共 储 A 储 兲 ,

共116兲

where k and ␻ i satisfy the time-asymptotic Vlasov dispersion relation, Eq. (111), and the amplitudes ␺ k, ␻ i satisfy the bifurcation equation Q KM 共 ⌿⫹⌽ 共 ⌿ 兲兲 ⫽0,

共117兲

where ⌽(⌿)⫽o( 储 A 储 ) is determined by the auxiliary equation, Eq. (115). Proof: In the light of the previous results, the proof reduces to establishing Eq. 共116兲. For any pair (k, ␻ i ) such that D(k, ␻ i )⫽0, Eq. 共115兲 gives

␾ k, ␻ i ⫽ 关 D共 k, ␻ i 兲兴 ⫺1 Mk, ␻ i 共 A⫹T 0 ,F ␣ ,h ␣ 兲 ,

共118兲

Mk, ␻ i 共 A⫹T 0 ,F ␣ ,h ␣ 兲 ⫽Nk, ␻ i 共 A⫹T 0 ,F ␣ ,h ␣ 兲 ⫺Lk, ␻ i 共 T 0 ,F ␣ ,h ␣ 兲 A.

共119兲

where

Mk, ␻ i (A⫹T 0 ,F ␣ ,h ␣ ) coincides with the expression on the first line in Eq. 共93兲 共since Nk, ␻ i (T 0 ,F ␣ ,h ␣ )⫽0 at the a.v.a.s.兲 and 兩 ␾ k, ␻ i 兩 ⭐ 兩 D共 k, ␻ i 兲 ⫺1 兩兩 Mk, ␻ i 共 A⫹T 0 ,F ␣ ,h ␣ 兲 兩

⭐ 兩 D共 k, ␻ i 兲 ⫺1 兩 储 A 储储 E共 k, ␻ i 兲 共 T 0 ⫹A,F ␣ ,h ␣ 兲 ⫺E共 k, ␻ i 兲 共 T 0 ,F ␣ ,h ␣ 兲储 ⫽o 共 储 A 储 兲 .

共120兲

The decomposition a⫽⌿⫹⌽ corresponds to the decomposition of the time-asymptotic field A in the form A 共 x,t 兲 ⫽A K共 x,t 兲 ⫹A H共 x,t 兲 ,

共121兲

where A K共 x,t 兲 ⬃

兺

k, ␻ i

␺ k, ␻ i e ikx⫹i ␻ i t ,

A H共 x,t 兲 ⬃

兺

k, ␻ i

␾ k, ␻ i e ikx⫹i ␻ i t .

共122兲

Since these Fourier–Bohr series can be summed uniformly 共e.g., via the Fejer–Bochner summation method兲 Eq. 共120兲 implies that 储 A H储 ⫽o 共 储 A 储 兲

共123兲

and Eq. 共116兲 follows immediately. 䊐 As previously mentioned, in most cases of physical interest K is finite-dimensional, so that Eq. 共117兲 reduces to a finite system of N nonlinear algebraic equations for the amplitudes ␺ k, ␻ i .

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C. Lancellotti and J. J. Dorning

VIII. CONCLUSION

The general solution for a small-amplitude time-asymptotic electric field near a critical a.v.a.s. is given by Eq. 共116兲 as a superposition of undamped traveling-wave modes, whose frequencies and wave numbers satisfy the time-asymptotic Vlasov dispersion relation, Eq. 共111兲. Remarkably, this result holds regardless of the details of the transient field T. Of course, T determines the T time-asymptotic Vlasov equilibrium F ␣0 , and also the amplitudes a k, ␻ i via the nonlinear operator M in Eqs. 共115兲 and 共117兲. In this sense, the results reported here represent the first step in the study of the initial value problem for the VPA system near a critical a.v.a.s.; a complete quantitative analysis requires the study of the transient equation, Eq. 共22b兲, and then of the bifurcation equation, Eq. 共117兲. An example of how this can be done has recently been given by the authors in Ref. 20 共see also Ref. 14兲 where they derived an approximate solution for f ␣ from a ‘‘transiently linearized’’ Vlasov equation. That equation was solved analytically by applying Hamiltonian perturbation theory in order to determine the characteristics associated with a timeasymptotic field A of the form given by Eq. 共116兲. Here, however, we have developed a much more general framework for the rigorous analysis of the long-time behavior of waves propagating in plasmas. This framework relies on two essential results of the analysis we have reported. First, the fact that critical points for the VPA initial value problem are bifurcation points for the time-asymptotic equation opens an interesting new perspective on the study of nonlinear Landau damping, and in particular on the transition between initial conditions that Landau damp to a zero electric field and those that lead to a nonzero time-asymptotic field.20 Second, the determination of the time-asymptotic linearized operator Lk, ␻ i provides a new and more solid foundation for the nonlinear analysis of the VPA system. By computing the Fourier– Bohr coefficients Nk, ␻ i in the form of Eq. 共88兲, we have been able, in effect, to take the limit t ˜⬁ before linearizing the nonlinear Ampe`re equation. As a consequence, we have obtained a linear approximation to the time-asymptotic VPA problem which is very different from the traditional linear theory6,25,26 and which is uniformly valid in the time-asymptotic limit. Indeed, the linearization is not about an initial unperturbed Vlasov equilibrium F ␣ , but about the timeT asymptotic Vlasov equilibrium F ␣0 , that incorporates the cumulative effects of the transient field T 0 . It is very significant that the corresponding nonlinear solution for the time-asymptotic electric field, Eq. 共115兲, is formally so similar 共especially in terms of the Vlasov dispersion relation兲 to the multiple-wave BGK-type nonlinear solutions12 which have been traditionally regarded as the natural candidates to describe long-time plasma-wave propagation. C. Bardos and P. Degond, Ann. Inst. Henri Poincare`, Analyse Nonline´aire 2, 101 共1985兲. J. Cooper, Math. Methods Appl. Sci. 5, 516 共1983兲. 3 R. Di Perna and P. L. Lions, C. R. Acad. Sci. Paris Se´r. I Math. 307, 655 共1988兲. 4 K. Pfaffelmoser, J. Diff. Eqns. 95, 281 共1992兲. 5 T. M. O’Neil, Phys. Fluids 8, 2255 共1965兲. 6 L. D. Landau, J. Phys. 共U.S.S.R.兲, 10, 25 共1946兲. 7 J. H. Malmberg and C. B. Wharton, Phys. Rev. Lett. 19, 775 共1967兲. 8 L. Demeio and P. F. Zweifel, Phys. Fluids B 2, 1252 共1990兲. 9 G. Manfredi, Phys. Rev. Lett. 79, 2815 共1997兲. 10 J. P. Holloway and J. J. Dorning, Phys. Rev. A 44, 3856 共1991兲. 11 M. L. Buchanan and J. J. Dorning, Phys. Rev. E 52, 3015 共1995兲. 12 M. Buchanan and J. J. Dorning, Phys. Rev. Lett. 70, 3732 共1993兲; Phys. Rev. E 50, 1465 共1994兲. 13 I. B. Bernstein, J. M. Greene, and M. D. Kruskal, Phys. Rev. 108, 546 共1957兲. 14 C. Lancellotti, Ph.D. dissertation, University of Virginia, 1998. 15 A. Wintner, Am. J. Math. 67, 277 共1945兲. 16 M. Fre´chet, Rev. Sci. 79, 341 共1941兲. 17 A. M. Fink, Almost Periodic Differential Equations 共Springer, New York, 1974兲. 18 A. B. Rechester and T. H. Stix, Phys. Rev. A 19, 1656 共1979兲. 19 A. J. Klimas, J. Comput. Phys. 68, 202 共1987兲. 20 C. Lancellotti and J. J. Dorning, Phys. Rev. Lett. 81, 5137 共1998兲. 21 M. S. Berger, Nonlinearity and Functional Analysis 共Academic, New York, 1977兲. 22 F. Riesz and B. Sz.-Nagy, Functional Analysis 共Dover, New York, 1990兲. 23 A. Vlasov, J. Phys. 共U.S.S.R.兲, 9, 25 共1945兲. 1 2

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24

L. Cesari, in Lecture Notes in Pure and Applied Mathematics, Vol. 19 in Nonlinear Functional Analysis and Differential Equations, edited by L. Cesari, R. Kannan, and J. D. Schuur 共Marcel Dekker, New York, 1976兲. 25 N. G. Van Kampen, Physica 共Amsterdam兲 21, 949 共1955兲. 26 K. M. Case, Ann. Phys. 7, 349 共1959兲.

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