Debraj Mandal

Devendra Sharma

Hans Schamel

Ultra slow electron holes in collisionless plasmas: stability at high ion temperature

Numerical simulations recover ultra slow solitary electron holes (SEH) of electron-acoustic genre propagating stably well below the ion acoustic speed Cs where no pure electron perturbation is known to exist yet, as they are disallowed by the ion response. Recovered at high ion temperature (Ti > Te), the reason of this stability (unaccelerated propagation, unseen before in existing literature) of SEH is traced to the loss of neutralizing cold ion response. In the opposite case of a background of sufficiently cold ions, Te > 3.5Ti, SEHs are accompanied by an ion compression that yields phase velocities above Cs (ion acoustic genre) and accelerates them, forcing a jump over a forbidden velocity gap, and settle on the high velocity tail of the electron distribution fe. In the observed ultra slow structures having Ti > Te, however, the warm ions begin to supplement the electron response and show Boltzmann-like behavior, transforming the ion compression to decompression (rarefaction) at the hole location. SEHs hence belong to the continuous spectrum of slow electron acoustic-like modes being triggered by the electron trapping nonlinearity. The results also suggest a scope of generalization of the basic EH theory.

I. INTRODUCTION

The collective excitations in collisional plasmas are well represented by discrete linear waves below the amplitudes where the convective nonlinearity of fluid formulation begins to assume significance. In hot collisionless plasmas, however, the earliest (often vanishing) threshold to nonlinear behavior is introduced by the kinetic effects such that the waves very fast achieve coherency at unusually low amplitudes. The first accessible class of nonlinear collective excitations in hot plasmas therefore, in practice and in most numerical simulations, is that of the nonlinear particle trapping equilibria, such as the non-isothermal ion acoustic solitary waves, solitary and cnoidal electron and ion holes or various forms of double layers [1][2][3][START_REF] Krall | Principles of Plasma Physics[END_REF][START_REF] Bernstein | [END_REF][6][7]. Many of these nonlinear modifications have recently explained anomalous effects [8][9][10][11] and excitations in the circular particle beam (synchrotron), experiments [12][13][14].

We shall present here the first discovery of a solitary slow electron acoustic hole-type structure propagating stably below c s , if the ion temperature exceeds T e /3.5. For cold ions on the other hand previous studies have revealed accelerating holes that settle above c s . To identify the reported structures we consider that the simplest nonlinear analytic approach to the experimentally and numerically observable class of excitations [15][16][17][18][19][20][21][22] is based on a Maxwellian equilibrium distribution, as does the standard linear kinetic Vlasov analysis [33]. For the trapped electron distribution included in a nonlinear treatment, however, a variety of Ansätze is in principle possible but the choice of a single parameter analytic distribution for them would produce the simplest class of nonlinear solutions. Each nonlinear solution therefore becomes identifiable with a corresponding linear one.

We shall apply the Vlasov approach developed extensively by Schamel and co-authors [23] which intro-duces an amplitude dependence in the dispersion and removes much of the discreteness of the linear wave solution space. Describing from general viewpoint of the nonlinear Vlasov approach, it was recently discovered [24], that the discreteness of the linear modes (distinct roots of linear dispersion function, separated by bands) also gets reflected in the nonlinear solutions space (as the corresponding band gaps) which was originally understood to be continuum [23]. These band gaps, or forbidden velocity ranges, were identified to be admissible also by the EH theory after the simulations in [24] could achieve no stably propagating EH structures in particular velocity ranges. In more specific terms, the nonlinear EH structures were noted as unstable (i.e., not propagating coherently but accelerating) below a critical velocity value which additionally ruled out existence of any electron holes slower than nearly the ion acoustic speed, analytically reaffirming several past observations of only accelerating holes in the simulations [25][26][27] there.

In this paper we present a set of simulations showing that ultra slow electron holes regain their stability at large enough ion temperature which exceeds electron temperature. This remarkably shows that the slow electron acoustic-like excitations are capable of propagating with unconventionally slow velocities. The associated nonlinear analytics shows that the band gap is a dynamical one and may indeed be buried with the changing ion temperature, showing no minimum cutoff velocity (e.g., the ion acoustic speed) for structures with no ion trapping nonlinearity.

With no significant contribution of resonant ions and a decompressed electron density, the observed ultra slow EH correspond to the slow electron-acoustic structures. They however have an unusual ion density profile which is also decompressed, in contrast to the ion compression in their usual supersonic velocity regime. In the conclusion of this paper we finally highlight an important issue that the theoretical recovery of these slow electron hole structure under the basic EH theory may be possible only by a further extension of the theory. Although such an extension and greater details of this analytic aspect is addressed in a dedicated forthcoming article [START_REF] Schamel | 37MesoCentre[END_REF], the idea mainly pertains to phase-space topology of the trapped electron density of the observed ultra-slow stable electron hole structures summarized in following statements. While the observed slow EH are recovered to have a dip-like trapped electron density, the lowest order electron hole theory prescribes them to have exclusively a humped structure. A modification of the lowest order electron hole theory is however possible by appropriate higher order corrections, allowing it to yield a dip-like trapped electron density structure, as recovered numerically for these ultra slow structures, without causing any characteristic change in the associated pseudo-potential structure.

This paper is organized as follows. We present the results of our high resolution Vlasov simulations in Sec. II. The analytical model following the Vlasov formulation is discussed and used to describe the results in Sec. III. The discussion of the physics of the observation and requirement of appropriate extension of the EH theory is highlighted in Sec IV and the summary and conclusions are presented in Sec. V.

II. SIMULATION RESULTS

We performed Vlasov simulations using the Flux-Balance [31,32] technique for both electrons and ions in the x-v space with 8192 × 16384 dual mesh grid. A well localized initial perturbation is used in the electron distribution function with the analytic form of perturbation,

f 1 (x, v) = -ǫ sech v -v 1 L 1 sech 4 [k(x -x 1 )] ( 1
)
where ǫ is the amplitude of the perturbation, L 1 is the width of the perturbation in the velocity dimension and k -1 is its spatial width. The background equilibrium velocity distribution of the electrons is a shifted Maxwellian and that of the ions an unshifted Maxwellian, given in normalized quantities by

f 0e (v) = 1 √ 2π exp - (v -v D ) 2 2 ( 2
)
f 0i (u) = F 0 √ 2π exp - u 2 2 ( 3
)
where v is normalized by v the = Te me and u by v thi = Ti mi , respectively. Therefore u = v θ/δ, where θ = T e /T i and δ = m e /m i . Subscripts j = e, i correspond to electron and ion species, respectively. The factor F 0 is ratio of total perturbed and and unperturbed electron content in the simulation region ( f e dxdv/ (2π) -1/2 exp[-u 2 /2] dxdu), ensuring that the same number of ions and electrons are present in the simulation box. Note that F 0 approaches unity for perturbations having a vanishing phase space integral, e.g., a pure sinusoidal perturbation or in the limit of a small localized perturbation, like (1), in an infinitely long plasma. In the simulation we use the Debye length λ D , inverse electron plasma frequency ω -1 pe and electron thermal velocity v the as normalizations for length, time and electron velocities, respectively. According to linear theory of plasma, the critical linear threshold (v * D ), required minimum drift value v D for a current driven ion acoustic instability, becomes v * D = 0.053 for θ = T e /T i = 10 and δ = m e /m i = 1/1836. For all the cases our choice of the drift velocity is v D = 0.01 which is well below the linear threshold for those temperature ratios [START_REF] Krall | Principles of Plasma Physics[END_REF]33].

We first present the evolution of the total electron distribution f e = f 0e + f 1 in cases 1-5 having θ = 50, 30, 10, 1 and 0.1, as plotted in Fig. 1, respectively, showing result of varying ion response in them. Considering the temperature correction term, the ion acoustic wave phase velocity in one dimension for these five cases are C s = 0.033, 0.036, 0.045, 0.093, and 0.24v the , where

C s = ( √ δ + γ δ/θ) v the with γ = 3.
Therefore in all the cases the initial electron velocity perturbation location v 1 = 0.01 v the is well below the corresponding C s and also the drift velocity v D is well below the corresponding critical linear thresholds v * D . Moreover, in the last case the ion temperature is higher than the electron temperature. A nonlinear plasma response to the applied perturbation, in the form of amplitude dependent propagating coherent structures, is nevertheless seen in all the cases where the perturbation of the form ( 1) is placed at x = 15 in the simulation box having the length L = 30. The velocity perturbation location in all cases is v 1 = 0.01 with phase-space widths of the perturbation L 1 = 0.01 along the velocity dimension and k -1 = 10 along the spatial dimension x. The strength of the perturbation is small: ǫ = 0.02. As witnessed in our earlier simulations, being placed at such small velocity the initial perturbation with θ > 1 is unstable and experiences an acceleration. For the last case with θ = 0.1, however, the time evolution of the contours of electron distribution function f e (x, v) in phase-space, presented in last row (from top) of Fig. 1, shows that the perturbation is largely intact and, after a marginal readjustment of its x-v space widths, continues its propagation with nearly the original velocity, 0.01v the .

Considering insignificant contribution of resonant ions (a very narrow velocity range of ion trapping region, as compared to trapped electrons), the stability of electron holes for small θ is once again understood to be determined by collective shielding effects rather than resonant ion reflection [6]. In qualitative sense [24] it can be described as follows. In a stable hole, the flux of the cold ion density expelled by the positive potential of the perturbation balances the flux of ions pushed in by the rel-FIG. 1: Evolution of the electron phase-space perturbation in all the five cases. In case-5 (θ = 0.1) the electron hole is not accelerated from its initial perturbation location. In all other cases they accelerate but their evolution time is different. The cases with lower ion temperature (high θ) take more time to construct a valid electron hole solution from the initial perturbation. (The color scale is used for the value of electron distribution function, increasing linearly from blue to red.) ative excess of hot electrons surrounding the hole. Thus the stability is achieved at faster velocities because of smaller outflowing ion flux due to smaller exposure of background ions to the hole electric field [24]. The stability at smaller velocity therefore presents an interesting case and indicates a new mechanism underlying the stable holes to overcome destabilizing cold ion response that, in the usual case of colder ions, necessitates a minimum velocity for the stability. The slow holes observed in our simulation are found to achieve this stability by marginalizing the cold ion response in the limit θ ≪ 1. We observe that the stability is achieved critically when the single (fully untrapped) ion population stops supplementing the response of cold electrons and instead begins to supplement the response of streaming Boltzmann electrons. This means the warm ions rather rarefact at the hole location in full accordance with the Boltzmann-like response of positive ions to a positive potential. This behavior of ion density is clearly visible in the ion density profiles shown in Fig. 2 In the next section we examine this aspect more quantitatively and explain that these solutions are a special class of Cnoidal Electron Holes representing the nonlinear solutions of the Vlasov equation.

III. ANALYTICAL VLASOV MODEL OF ULTRA SLOW ELECTRON HOLE AND ITS STABLE EXISTENCE

The existence regimes of Cnoidal Electron Holes (CEHs) and their dependence on the ion temperature are now evaluated in more quantitative terms using the nonperturbative nonlinear dispersion formulation of Schamel (see e.g. [23] and references therein). Note that the description below is limited to identifying the limits that bound the parameter regime in which the formal solutions of Vlasov equation, prescribed in [23], represent an undamped propagation. A solution outside these does not satisfy nonlinear dispersion relation and therefore must undergo a transient, or phasemix, i.e. an evolution which is not covered by a nonlinear dispersion formulation that aims to identify only the coherently propagating solutions. A more detail description of the analytical model used here is given in Appendix-A. The phase velocity of a settled vortex structure in electron phase space is determined by the nonlinear dispersion relation (NDR) Eq. (A7),

k 2 0 - 1 2 Z ′ r ṽD / √ 2 - θ 2 Z ′ r u 0 / √ 2 = B, (4)
where, Z r (x) is the real part of the plasma dispersion function. Depending on different values of k 0 and B one gets different type of solutions, like solitary solution and cnoidal solutions. The left hand side of the equation presents the contribution of free electrons and ions and the right hand side presents the trapped electron contribution.

A. Underlying electron acoustic character of the SEH

The new understanding from the discovery of stability of ultra slow electron hole, in case V, is that a persistent dip in the electron density at the hole location is a result of the electron acoustic continuum which is cloaked by the ion response at the relatively colder ion temperatures. This robust electron acoustic physics becomes visible only in the limit θ → 0 where the streaming ion term (third term in LHS of the NDR (4)) becomes negligible and the structure is purely an equilibrium between trapped and streaming electron responses. As the most notable result of this study, the case V represents this limit and, moreover, it can be shown that this limit is equivalent to the limit of immovable ions, as recoverable using the ion density expression (A4) that yields n i = 1 for θ → 0 in case of no ion trapping.

Accordingly, the phase velocity of solitary EH continuum in the immobile ion limit (infinite T i ) is determined by the following reduced form of the NDR (4), containing only electron contributions,

- 1 2 Z ′ r ṽD / √ 2 = B, k 2 0 = 0.
Note that, from Fig. 3, 0 < ṽD < 0.924 √ 2 = 1.307 and only 0 < B ≤ 1 is admissible, providing exclusively electron hole-like solutions. It is solved for small B by ṽD = ±1.307(1 -B) or v 0 = v D ± 1.307(1 -B) and represents the continuous extension of the slow electron acoustic mode which is, as a discrete mode, defined by B = 0 and k 0 = 0 [2,7]. For B close to unity we instead have

v 0 = v D ± √ 1 -B.
In our subcritical regime of v D = 0.01 < v D * = 0.053 the SEH is hence placed on both sides of the electron distribution function very close to its top. Physically, while at low ion temperature, only the streaming electrons help to shield the electric field associated with the electron hole and the ions are compressed in the low electron pressure region of the EH. At high ion temperature, on the other hand, the hot ions also contribute to the shielding, essentially collaborating with streaming electrons. This explains the Boltzmann like density profile of the ions at θ ≪ 1 (case-V) in Fig. 2.

B. Gap of existence and the ion acoustic character

of the SEH at large θ

We now describe the limit θ ≫ 0 where ion contribution (third term in LHS of NDR ( 4)) can not be neglected. When ṽD → 0, additionally, one can simplify Eq. ( 4), by considering -1 2 Z ′ r ṽD / √ 2 ∼ 1 and k 2 0 = 0 for solitary electron hole, as

- 1 2 Z ′ r u 0 / √ 2 = 1 θ (B -1) =: D. ( 5
)
which describes SEH of ion acoustic genre since it produces, for small u 0 the value u 0 ≈ 1 -1 θ (B -1), i.e., for no nolinearity (B → 0) the structure travels with the pure ion acoustic velocity. A finite B (> 0 for SEH) can however allow deviation from this and produce a full ion acoustic velocity continuum. In general, three distinct regions from this continuum can be identified, based on the behavior of function -1 2 Z ′ r u 0 / √ 2 as plotted in Fig. 3,

(i) 0.0 ≤ u 0 ≤ 1.307 (= 0.924 √ 2), 1 ≤ B ≤ 1 + θ (ii) 1.307 < u 0 ≤ 2.12 (= 1.5 √ 2), 1 -0.285 θ ≤ B < 1 (iii) 2.12 < u 0 < ∞, 1 -0.285 θ ≤ B < 1.
Therefore, for cold ions θ > 3.5, the first two belong to the Slow Ion Acoustic branch (SIA), the third one to the ordinary Ion Acoustic branch (IA), which may have velocity greater than C s . In the second column the necessary conditions for B are presented for given θ, which

FIG. 3: Plot of -1 2 Z ′ r (x) with x.
are subject to B > 0. This means that for the doublet of solutions, (ii), (iii), which satisfy the same constraints, (-0.285 ≤ D < 0), there is a division line for θ given by θ * := 1 0.285 = 3.51. For cold ions, θ > 3.5, any B in 0 < B < 1 is admitted, whereas for hot ions, θ < 3.5, B must satisfy 1 -0.285 θ ≤ B < 1. In terms of B we have therefore the following situation, for slow regime of SIA given by (i) 1 ≤ B ≤ 1 + θ there is no other choice for a solution. But for 1 -0.285θ ≤ B < 1, the plasma has two choices for establishing a solution, u 0 lies either on the faster part of the SIA branch (ii) or on the still faster velocity IA branch, regime (iii), with a gap in between [24]. This gap region presents forbidden region for a stable electron hole solution of the ion acoustic genre.

Since ion acoustic velocity in the ion frame is given by C s := √ θ for θ values, 0.1, 1 and 10 we get the triplet C s = (0.32, 1, 3.16), and the associated Mach numbers M := u0 cs the triplet (0.43, 1, 1.5). The SEH structure hence travels subsonically for hot and supersonically for cold ions, whereas it moves sonically for moderate ion temperatures, θ ≈ 1. Moreover, for θ = 10 (and 30,50) the SEH, originated in region (ii), makes a transition from (ii) to (iii), i.e., it accelerates and jumps the above gap, of solution ("forbidden region"), to settle in the supersonic regime [24].

With respect to the pure kinetic effects of ions reflection as treated by Dupree [6] not significantly visible in present cases [24], the simulations highlight the dominant role of streaming ion population in comparison to the reflected ion population duly accounted for in the present simulations. Note that the width, along the velocity dimension, of the ion trapping region is smaller by a factor θ/δ in comparison to that of electrons because of higher kinetic energy of ions at similar velocities. In other words, a small amplitude structure would not trap/reflect as large fraction of ions density as that of electrons. Although this small reflected ion population effectively represents a trapped ion population in our periodic setup, it does not maintain its identity, distinct form the streaming population, over its longer transit between two consecutive reflections (more so in the limit L → ∞). This justifies neglecting the ion trapping term b(α, u 0 ), as in the NDR Eq. (A7), since reflected/trapped ions may not effectively maintain an α value different from the unity. Moreover, for present EH having small ψ << T e the net momentum transferred, because of finite ∂f i /∂v, by the imbalanced populations of reflected ions to the hole, as considered by Dupree [6], is negligible given the narrow width of ion trapping region along the velocity dimension, as discussed above. Therefore, the response of streaming ions remains the most dominant factor in determining the observed stability of the hole solutions, as considered in the present analysis.

We close this theoretical part with a few experimentally relevant remarks on spontaneous acceleration of holes. A similar sudden acceleration of holes (in this case of a periodic train of ion holes) has been seen in the experiments of [35]. Density fluctuation measurements in a double plasma device show an apparently spontaneous transition of these periodic structures from slow ion acoustic to ion acoustic velocity regime. In this experiment gradual scattering of the trapped ion population by elastic collisions with neutrals was made responsible for this transition. The outcome of our paper, however, suggests a second possibility as an alternative explanation, namely the tendency of the plasma to achieve a lower energy status during the evolution, a process which will be the more probable the more dilute the plasma is.

IV. MISSING HUMP AND EXTENSION OF THE BASIC EH THEORY

We now indicate an advanced feature of the Electron hole solutions identified in the present simulation output which might require extension of the basic EH theory to include a newer parameter to accommodate multiplicity of scales in trapped electron density.

Note that the solution in Sec. III are discussed under the approximation k 2 0 = 0, appropriate for a solitary EH with depressed trapped electron electron density. However, when we examine the numerically recovered values of quantity k carefully, the basic EH theory for these numerical k values prescribes that for small θ solution the electron density must feature a hump like profile. The plasma, however, avoids this less stable state by transiting to a multiple-scale state of the trapped density where the central phase-space density of the trapped electrons still features a sharp dip, surrounded by a relatively flatter density profiles. Quantitatively, this situation is resolvable only by introducing more sophistication in the hole theory, which is a topic addressed in a forthcoming article dedicated to this issue and such a generalizing modification of the EH theory [START_REF] Schamel | 37MesoCentre[END_REF].

V. SUMMARY AND CONCLUSIONS

In summary, we have proved numerically and theoretically the stable existence of hole solutions in subcritical plasmas occurring at very high ion temperature values. This outcome is striking as it manifests the electron trapping nonlinearity as the ruling agent in this evolutionary process standing outside the realm of linear wave theory. We could show that the potential φ(x) in SEH is essentially a local property of the resonant electrons in phase space (via β) whereas, the dynamics or hole speed is governed (a) in hot ion regime by streaming electrons and an almost immobile-like ion presence, and (b) in cold ion regime by an optimum of electron shielding (- 1 2 Z ′ r ṽD / √ 2 = 1) and a variable, T i -dependent, ion shielding (or compression). We illustrated that it is the ion response which "destabilizes" the electron hole structure when T i < T e /3.5, causing the slower hole to accelerate, to jump over a forbidden velocity interval and to approach a higher speed settling in the high velocity wing of the electron distribution at a lower energy plasma state. For higher ion temperatures T i > T e /3.5 these slow holes have already achieved this status by a reversal of ion shielding that marginalizes the cold ion response. Related to phase-space topology of the trapped electrons in ultra slow EH the present simulation has importantly indicated that in order to model the observed density dip involving trapping with multiple scales, a further modification of the basic EH theory would be necessary. The same would be subject of a forthcoming article [START_REF] Schamel | 37MesoCentre[END_REF]. Note that the approximate analytical limiting value T e /3.5 for existence of coherent solutions may also be subject to modification in any further improved formulation.

Our description utilizes the Vlasov equation in its full nonlinear version rather than the truncated linear Vlasov version. The observed structures and corresponding quantitative analysis presented provide the foundation for treating the mechanism of nonlinear stability in a number of conditions of high physical relevance where ion temperature either approaches or exceeds the electron temperature. The explanation of parallel activity in low frequency turbulence phenomena in the edge of magnetized fusion plasmas can be further supplemented by such slow structures. The drift-wave turbulence remains the basic model for the perpendicular activity in such magnetized conditions. While stellar or magnetospheric plasmas with nonthermal species are prime candidates, hot plasmas where T i exceeds T e with hole structures caused by edge turbulence may be found in the edge of International Thermonuclear Experiment Reactor (ITER) [START_REF] Drive | ITER Physics Expert Group on Energetic Particles[END_REF], or in the desired operating limit of the transport in current free core plasmas of helical confinement devices like LHD [29] and in modern stellarators like W7-X [30].

1 - 1 FIG. 2 :

 112 FIG.2: Late time plots of Electron density (red), ion density (blue) and potential (magenta) profiles for (a) θ=30 and (b) θ=0.1. The dominant positive value peak present in the potential profiles (at the location of dotted line) for both the cases is the excited EH structure. The electron density has a coinciding sharp dip in both the cases at EH location whereas the ion density features, a peak in for cold ion case (a) but a dip for the hotter ion case (b).

  features a peak in for cold ion case (a) but a dip for the hotter ion case (b). The additional structures upstream of EH in (a) are product of the transient during accelerating phase (appearing ahead of it because of the periodicity), which remains absent in (b)

Acknowledgements

We are grateful to Professors Yves Elskens, Dominique Escande and Fabrice Doveil for many fruitful discussions and their comments, and anonymous reviewers for constructive remarks. This work was granted access to the HPC resources of Aix-Marseille Université [37] financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program Investissements d'Avenir supervised by the Agence Nationale de la Recherche.

Appendix A: Analytical Model of solitary electron hole (SEH)

The analytic expressions of the electron and ion distribution function for a solitary electron hole (SEH) solution in presence of an electron current in a Vlasov Plasma system, are given by H. Schamel [23]

(A1)

Where,

x is normalized to λ De = T e /4πn 2 e and v is normalized by the electron thermal velocity, v the := T e /m e , u is normalized by ion thermal velocity, u thi := T i /m i . The relation between u and v is, u = µv. Here, µ = (m i T e /m e T i ) 1/2 and v 0 and u 0 are the phase velocity of the wave in the electron and ion frame, respectively. ṽD = v D -v 0 , where v D is the drift velocity of the electrons. K = k 2 0 ψ/2 is related to the wave number. k 2 0 > 0 and k 2 0 = 0 correspond to a periodic (cnoidal) wave and a solitary wave solution, respectively. A e is a constant. θ := T e /T i and α , β are the trapping parameters, respectively. Velocity integration of the Eq. ( A1) and (A2) yields in the small amplitude limit, ψ ≪ 1, for solitary waves:

where b(β, ṽD ) and b(α, u 0 ) determine the trapped particle density of electron and ion, respectively. In the absence of ion trapping nonlinearity b(α, u 0 ) = 0 and b(β, ṽD ) is given by:

In the Eq. ( A3) and (A4) Z r (x) = -2e -x 2 x 0 dt exp (t 2 ) is the real part of the plasma dispersion function and φ is the potential, satisfying the Poisson equation.

The pseudo potential V (φ) associated with the Poisson equation (A5) is given by:

The phase velocity of the structure is determined through the Nonlinear Dispersion Relation (NDR) given in terms of β and k 2 0

We define B := 16 15 b(β, ṽD )ψ 1/2 . Substituting (A6) in (A5) and subsequent integration leads in the limit k 2 0 → 0, applicable to the existence of a solitary wave solution, to :

which requires a positive B, B > 0. The expression (A8) is independent of θ and holds for any θ in 0 ≤ θ ≤ ∞, whereas the phase velocities v 0 , u 0 follow from (A7) and the relation u 0 = θ δ v 0 , as discussed in the paper.