Nonlinear wave-wave interactions in quantum plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Nonlinear wave-wave interactions in quantum plasmas
A. P. Misra
1, 2, ∗ and P. K. Shukla
1, 3, † Department of Physics, Ume˚a University, SE–901 87 Ume˚a, Sweden. Department of Mathematics, Visva-Bharati University, Santiniketan-731 235, India. RUB International Chair, International Centre for Advanced Studies in Physical Sciences,Faculty of Physics & Astronomy, Ruhr University Bochum, D-44780 Bochum, Germany. (Dated: 07 September, 2010)Wave-wave interaction in plasmas is a topic of important research since the 16th century. Theformation of Langmuir solitons through the coupling of high-frequency (hf) Langmuir and low-frequency (lf) ion-acoustic waves, is one of the most interesting features in the context of turbulencein modern plasma physics. Moreover, quantum plasmas, which are ubiquitous in ultrasmall elec-tronic devices, micromechanical systems as well as in dense astrophysical environments are a topicof current research. In the light of notable interests in such quantum plasmas, we present here atheoretical investigation on the nonlinear interaction of quantum Langmuir waves (QLWs) and quan-tum ion-acoustic waves (QIAWs), which are governed by the one-dimensional quantum Zakharovequations (QZEs). It is shown that a transition to spatiotemporal chaos (STC) occurs when thelength scale of excitation of linear modes is larger than that of the most unstable ones. Such lengthscale is, however, to be larger (compared to the classical one) in presence of the quantum tunnelingeffect. The latter induces strong QIAW emission leading to the occurrence of collision and fusionamong the patterns at an earlier time than the classical case. Moreover, numerical simulation of theQZEs reveals that many solitary patterns can be excited and saturated through the modulationalinstability (MI) of unstable harmonic modes. In a longer time, these solitons are seen to appear inthe state of STC due to strong QIAW emission as well as by the collision and fusion in stochasticmotion. The energy in the system is thus strongly redistributed, which may switch on the onset ofLangmuir turbulence in quantum plasmas.
PACS numbers: 52.25.Gj, 52.35.Mw, 05.45.MtKeywords: Quantum Zakharov equations, Solitary patterns, Spatiotemporal chaos
INTRODUCTION
Quantum Zakharov equations (QZEs) [1] describe the nonlinear interaction of high-frequency (hf) quantum Lang-muir waves (QLWs) and the low-frequency (lf) quantum ion-acoustic waves (QIAWs) in quantum plasmas. This set ofequations basically extends the classical Zakharov equations (CZEs) [2] with higher-order dispersive terms associatedwith the Bohm potential. Unlike CZEs, QZEs have been deduced using a quantum fluid model under the similarquasineutral assumption and the multiple time scale technique. Recently, much attention has been paid to investigatethe dynamics of QZEs in the context of e.g., the formation of Langmuir solitons through modulational instability(MI) as well as Langmuir turbulence through the process of chaos (See, e.g., Refs. [3–7]). The statistical properties ofthe QZEs have been analyzed using kinetic treatment and to show that the quantum coupling parameter ( H ) can beresponsible for reducing the MI growth rate [8]. The latter is , however, shown to be maximized in the limit of H = 0.A variational approach was conducted to study the quantum effects on localized solitary structures [3]. Moreover,some general periodic solution using Lie point symmetries [4], some exact solutions [5] as well as both temporal [6] andspatiotemporal dynamics [7] in the context of chaos and Langmuir turbulence of the QZEs have been investigatedin the recent past. Furthermore, a comprehensive work on the dynamics of Langmuir wave packets in three spatialdimensions [9] as well as some investigations on arrest of Langmuir wave collapse [10] by the quantum effects can befound in the recent works.Notice that when the wave electric field is strong such that it approaches the decay instability threshold, theinteraction of QLWs and the QIAWs is said to be in ‘weak turbulence’, and then QLWs are essentially scatteredoff QIAWs. On the other hand, when the electric field intensity is so strong that the MI threshold is exceeded, theinteraction results in ‘strong turbulence’ regime in which QLWs are typically trapped by the density cavities associated ∗ Electronic address: [email protected] † Electronic address: [email protected] with the QIAWs. Such phenomena can frequently occur in plasmas. Moreover, the transfer of energy to few strongermodes with small spatial scales can take place due to a chaotic process, and this energy transfer can be faster whenthe chaotic process in a subsystem of the QZEs is well developed [6].In the present work, we will investigate the full QZEs numerically especially when the wave number of modulation issmall enough from its critical value. The latter, in turn, excites many unstable harmonic modes which in a longer timecollide and fuse into fewer new incoherent patterns due to QIAW emission. We will show that since the critical wavenumber of modulation (below which the MI sets in) depends on the quantum coupling parameter H, the length scale ofexcitation for the transition from temporal chaos (TC) to spatiotemporal chaos (STC) is to be larger than the classicalcase. Moreover, lower the values of H <
1, the higher is the number of unstable harmonic modes. Furthermore, wewill show that the solitary waves thus formed due to MI will lose their strength after a long time through randomcollision and fusion among the patterns under strong QIAW emission. This process becomes quicker whenever thedensity correlation due to quantum fluctuation becomes strong with higher values of the quantum parameter H . TheSTC state is then said to emerge, and the energy of the system is thus redistributed to new incoherent patterns aswell as to few stronger modes with small length scales. SPATIOTEMPORAL EVOLUTION OF QZES
The nonlinear interaction of QLWs and QIAWs is described by the following one-dimensional QZEs [1]. i ∂E∂t + ∂ E∂x − H ∂ E∂x = n E , (1) ∂ n∂t − ∂ n∂x + H ∂ n∂x = ∂ | E | ∂x , (2)where E = E ( x, t ) is the slowly varying wave envelope of the hf electric field and n = n ( x, t ) is the lf plasma densityperturbation due to QIAW fluctuation. Also, H = ~ ω i /k B T e , which is associated with the Bohm potential, representsthe ratio of the ion plasmon energy to the electron thermal energy. Here ~ is the scaled Planck’s constant, k B isthe Boltzmann constant, T e is the electron temperature and ω i ( e ) = p n e /m i ( e ) ε is the ion (electron) plasmafrequency with n denoting the constant background density and m i ( e ) the ion (electron) mass. The electric field E is normalized by p m e n k B T e /m i ε , the density n by 4 m e n /m i . Moreover, the space and time variables arerescaled by ( λ e / p m i /m e and m i / m e ω e , where λ e is the electron Debye length. Note that by disregarding theterm ∝ H in Eqs. (1) and (2), one recovers the well-known CZEs [2]. The latter have been widely studied in thecontext of solitons, chaos and Langmuir turbulence in many areas of plasma physics (see, e.g., Refs. [11–13]). It isthus of natural interest to investigate the QZEs in the quantum realm, which may be useful for understanding theonset of plasma wave turbulence at nanoscales in both laboratory and astrophysical plasmas.The growth rate of MI can be obtained from Eqs. (1) and (2) by assuming the perturbations of the form exp( ikx − iωt ) from a spatially homogeneous pump electric field E as [8]Γ = 1 √ h ̥ k p ̥ + 8 | E | − ̥ k (2 − ̥ k ) − ̥ k (1 + ̥ k ) i / , (3)where ̥ = 1 + H k and k < √ E / ̥ . It can be shown that the growth rate is maximum at H = 0 and decreasesfor increasing values of H with cut-offs at lower wave numbers of modulation [8].In order to solve numerically the Eqs. (1) and (2) we choose the following initial condition [7, 13] E ( x,
0) = E [1 + β cos( kx )] , n ( x,
0) = −√ E kβ cos( kx ) , (4)where E is the amplitude of the pump Langmuir wave field and β is a constant of the order of 10 − to emphasizethat the perturbation is very small. We use Runge-Kutta scheme with time step size dt = 0 . x = 0 corresponds to the grid position 1024. The spatial derivatives are approximated withcentered second-order difference approximations. The results are presented in Figs. 1-6 after the end of the simulationwith t = 200 and E = 2 . Note that as above the MI sets in for wave numbers satisfying 0 < k < k c , where k c is a −20 −15 −10 −5 0 5 10 15 200510 x | E ( x ,t ) | −20 −15 −10 −5 0 5 10 15 20−150−100−500 x n ( x ,t ) FIG. 1: (Color online) The profiles of the wave electric field E ( x, t ) (upper panel) and the associated density fluctuation n ( x, t )(lower panel) with respect to the space x after time t = 200 in the numerical simulation of the QZEs for k = 0 . H = 0 and E = 2.FIG. 2: (Color online) Space-time contour plots of | E ( x, t ) | =constant corresponding to the evolution as in Fig. 1. This showsthat pattern selection leads to many solitary patterns which due to QIAW emission collide and get fused into few incoherentpatterns after a long time. The STC state emerges. real root of the cubic H k + k − √ E = 0 , and k = k c defines the curve along which pitchfork bifurcation takesplace. Moreover, the dynamics is subsonic in the regime k c / < k < k c where the MI growth rate is small. As k is lowered from k c / , many unstable modes with higher harmonic modes will be excited. Again, the master modecan, in principle, result in the excitation of N − N = [¯ k − ] with ¯ k = k/ √ E . There may also exist many solitary patterns with spatially modulated length l m = L/m, where m = 1 is for mastermode ( l = L = 2 π/k ) and m = 2 , , ...., M are for the unstable harmonic modes. As a result, the envelope E can beexpressed as: −60 −40 −20 0 20 40 60024681012 x | E ( x ,t ) | −60 −40 −20 0 20 40 60−100−500 x n ( x ,t ) FIG. 3: (Color online) The same as in Fig. 1, but for H = 0 . k = 0 .
048 and other parameters remain the same as in Fig. 1. E ( x, t ) = M X m =1 E m ( t ) exp( im ¯ kx ) + ∞ X m = M +1 E m ( t ) exp( im ¯ kx ) , (5)in which the first term on the right-side of Eq. (5) comes from the master mode and unstable harmonic modes with M < N − k = 0 . H = 0. We observe an excited electric field of the order | E | ∼
12 highly correlated with density depletion n ∼ −
152 or n/n ∼ − . . From Fig. 2, we find that many solitary patterns are formed from the master mode andunstable harmonic modes by means of pattern selection. Two solitary patterns initially peaked in 100 < x < x ∼
50 and fuses into a newstrong mode. After few more other collisions and fusions, there remain only four distorted patterns. The system isthen said to emerge the STC state. When k is further lowered many solitary trains are excited and saturated. Assoon as the collisions and fusions among the solitary patterns take place and the new incoherent trains are formedunder strong QIAW emission, these incoherent patterns then collide with some others repeatedly and finally thereremain few incoherent patterns in stochastic motion with the greater part of the system energy. Meanwhile manystronger modes are also excited to share the energy. These are shown in Figs. 3, 4 and Figs. 5, 6 for different valuesof H : H = 0 . . H (Fig.6) the collisions and fusions take place at an earlier time than the case of lower H (Fig.4). Thus, a certain amountof energy which was initially distributed among many solitary waves, will now be transferred to a few incoherentpatterns as well as to some stable higher harmonic modes with short wave lengths. So, if initially there form manysolitary pattern trains due to MI with different modulational lengths, collision and fusion among most them can leadto the STC state. There must exist a critical wavelength or wave number k, at which the transition from temporalchaos to STC takes place. This value, in the quantum case ( H = 0) is quite different from the classical one ( H = 0).For detail investigations readers are referred to works, e.g., in Refs. [7, 13]. FIG. 4: (Color online) Contours of | E ( x, t ) | =constant with respect to space and time corresponding to the evolution as inFig. 3. This shows that pattern selection leads to many more solitary patterns than the classical case (Fig. 2) which due toQIAW emission collide and get fused into few incoherent patterns after a long time. The collision is random and not confinedbetween two patterns. Few stable modes are also excited. The system energy is then redistributed to new incoherent patternsand few stable modes with small length scales. The system is in the state of STC. −60 −40 −20 0 20 40 6002468 x | E ( x ,t ) | −60 −40 −20 0 20 40 60−40−30−20−10010 x n ( x ,t ) FIG. 5: (Color online) The same as in Fig. 3, but for different H = 0 .
5. Other parameters remain the same as in Fig. 3.
CONCLUSION
We have performed a simulation study of the QZEs to show that many coherent solitary patterns can be excitedand saturated through the MI of unstable harmonic modes by a modulation wave number of QLWs. It is observedthat there exist critical values of k for which the motion of the coherent solitary patterns is in the state of TC or inSTC. The transition from TC to STC occurs when k . .
14 for H = 0 or, for k . .
048 when H & . . It is shownthat the dispersion due to quantum tunneling induces strong QIAW emission leading to collision and fusion among
FIG. 6: (Color online) Contours of | E ( x, t ) | =constant with respect to space and time corresponding to the evolution as inFig. 5. Here the pattern selection leads to many solitary patterns, but less than that in Fig. 4. Though, the collision is randomand not confined between two patterns, but they fuse into new incoherent patterns in a shorter time than the case of H = 0 . the patterns to occur at an earlier time than the classical case. The Collision and fusion among some trains take placeand the new incoherent pattern trains are formed accompanying strong QIAW emission due to quantum effects. TheSTC state is then said to emerge. As a result, the system energy in the STC state is spatially redistributed in theprocess of pattern collision, fusion and distortion, which may switch on the onset of Langmuir turbulence in quantumplasmas. ACKNOWLEDGMENTS
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