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Plasma waves are trapped in regions of low density becausetheir dispersion relation32ω2 ¼ ω2p þ k2 v2thð4:30Þpermits waves of large k to exist only in regions of small ωp. The trapping of part ofthe k spectrum further enhances the wave intensity in the regions where it wasalready high, thus causing the envelope to develop a growing ripple.The reason the sign of pq matters is that p and q for plasma waves turn out to beproportional, respectively, to the group dispersion dvg/dk and the nonlinear frequency shift δω / ∂ω/∂jψj2. We shall show later that3148 Nonlinear EffectsFig.
8.27 Modulationalinstability occurs when thenonlinear frequency shiftand the group velocitydispersion have oppositesignsp¼1 dvg2 dkq¼∂ω∂jψ j2/ δωð8:123ÞModulational instability occurs when pq > 0; that is, when δω and dvg/dk haveopposite signs. Figure 8.27 illustrates why this is so. In Fig.
8.27a, a ripple in thewave envelope has developed as a result of random fluctuations. Suppose δω isnegative. Then the phase velocity ω/k, which is proportional to ω, becomes somewhat smaller in the region of high intensity. This causes the wave crests to pile up onthe left of Fig. 8.27b and to spread out on the right. The local value of k is thereforelarge on the left and small on the right. If dvg/dk is positive, the group velocity will belarger on the left than the right, so the wave energy will pile up into a smaller space.Thus, the ripple in the envelope will become narrower and larger, as in Fig. 8.27c. Ifδω and dvg/dk were of the same sign, this modulational instability would not happen.Although plane wave solutions to Eq.
(8.123) are modulationally unstable whenpq > 0, there can be solitary structures called envelope solitons which are stable.These are generated from the basic solution" # 1=22AA 1=2wðx; tÞ ¼sechx eiAtð8:124Þqp8.8 Equations of Nonlinear Plasma Physics315Fig. 8.28 An envelopesolitonwhere A is an arbitrary constant which ties together the amplitude, width, andfrequency of the packet.
At any given time, the disturbance resembles a simplesoliton (Eq. (8.100)) (though the hyperbolic secant is not squared here), but theexponential factor makes w(x, t) oscillate between positive and negative values. Anenvelope soliton moving with a velocity V has the more general form (Fig. 8.28)ψ ðx; tÞ ¼2Aq1=2" #A 1=2VV2sechðx x0 VtÞ exp i At þ x t þ θ0p2p4pð8:125Þwhere x0 and θ0 are the initial position and phase. It is seen that the magnitude ofV also controls the number of wavelengths inside the envelope at any given time.Problems8.19 Show by direct substitution that Eq. (8.124) is a solution of Eq.
(8.122).8.20 Verify Eq. (8.125) by showing that, if w(x, t) is a solution of Eq. (8.122), thenVV2ψ ¼ wðx x0 Vt, tÞexp ix t þ θ02p4pis also a solution.We next wish to show that the nonlinear Schr€odinger equation describes largeamplitude electron plasma waves. The procedure is to solve self-consistently for thedensity cavity that the waves dig by means of their ponderomotive force and for thebehavior of the waves in such a cavity. The high-frequency motion of the electronsis governed by Eqs.
(4.28), (4.18), and (4.19), which we rewrite respectively in onedimension as∂ue3KT e ∂n¼ E∂tmmn0 ∂xð8:126Þ∂n∂uþ n0¼0∂t∂xð8:127Þ3168 Nonlinear Effects∂E¼ ε10 en∂xð8:128Þwhere n0 is the uniform unperturbed density; and E, n, and u are, respectively, theperturbations in electric field, electron density, and fluid velocity. These equationsare linearized, so that nonlinearities due to the u · ∇u and ∇ · (nu) terms are notconsidered. Taking the time derivative of Eq. (8.127) and the x derivative ofEq.
(8.126), we can eliminate u and E with the help of Eq. (8.128) to obtain22∂ n 3KT e ∂ n n0 e2þn¼0∂t2m ∂x2 mε0ð8:129ÞWe now replace n0 by n0 + δn to describe the density cavity; this is the onlynonlinear effect considered. Equation (8.129) is of course followed by any of thelinear variables. It will be convenient to write it in terms of u and use the definitionof ωp; thus22∂ u 3KT e ∂ uδn2þ ωp 1 þu¼0m ∂x2∂t2n0ð8:130ÞThe velocity u consists of a high-frequency part oscillating at ω0 (essentially theplasma frequency) and a low-frequency part ul describing the quasineutral motionof electrons following the ions as they move to form the density cavity.
Both fastand slow spatial variations are included in ul.Letuðx; tÞ ¼ ul ðx; tÞeiω0 tð8:131ÞDifferentiating twice in time, we obtain2∂ u ¼ €ul 2iω0 u_ l ω20 ul eiω0 t2∂twhere the dot stands for a time derivative on the slow time scale. We may thereforeneglect u€l, which is much smaller than ω20 ul:2∂ u¼ ω20 ul þ 2iω0 u_ l eiω0 t2∂tSubstituting into Eq. (8.130) gives" #23KT e ∂ ul222 δn2iω0 u_ l þþ ω0 ω p ω pul eiω0 t ¼ 0n0m ∂x2We now transform to the dimensionless variablesð8:133Þ8.8 Equations of Nonlinear Plasma Physics031700t ¼ ω ptω ¼ ω=ω px ¼ x=λD001=2u ¼ uðKT e =mÞδn ¼ δn=n0ð8:134Þobtaining"#02 0 00 0∂ul 3 ∂ ul 1 0 20iω0 0 þþ ω 0 1 δn ul eiω0 t ¼ 02 ∂x0 2 2∂t0Defining the frequency shift Δ0Δ ω0 ω p =ω p ¼ ω0 1ð8:135Þ02and assuming Δ 1, we have ω 0 1 2Δ.
We may now drop the primes (these0being understood), convert back to u(x, t) via Eq. (8.131), and approximate ω0 by1 in the first term to obtain2i∂u 3 ∂ u 1þδnu¼0þΔ2∂t 2 ∂x2ð8:136ÞHere it is understood that ∂/∂t is the time derivative on the slow time scale,although u contains both the exp (iω0t) factor and the slowly varying coefficientul. We have essentially derived the nonlinear Schr€odinger equation (8.122), but itremains to evaluate δn in terms of julj2.The low-frequency equation of motion for the electrons is obtained byneglecting the inertia term in Eq.
(4.28) and adding a ponderomotive force termfrom Eq. (8.44)0 ¼ enE KT e∂n ω2p ∂ ε0 E2 2:∂x ω0 ∂x 2ð8:137ÞHere we have set γ e ¼ 1 since the low-frequency motion should be isothermal ratherthan adiabatic. We may set 2 m2 ω20 2 E ffi 2 ueð8:138Þby solving the high-frequency Eq. (8.126) without the thermal correction. WithE ¼ ∇ϕ and χ ¼ eϕ/KTe, Eq. (8.137) becomes∂1 m ∂ 2ðχ ln nÞ u ¼02 KT e ∂x∂xð8:139Þ3188 Nonlinear EffectsIntegrating, setting n ¼ n0 + δn, and using the dimensionless units Eq. (8.134), wehave1 2u214¼ juj2 ¼ χ lnð1 þ δnÞ ffi χ δnð8:140ÞWe must now eliminate χ by solving the cold-ion Eqs.
(8.103) and (8.104). Sincewe are now using the electron variables Eq. (8.134), and since Ωp ¼ Eωp,vs ¼ E(KTe/m)1/2, where E (m/M )1/2, the dimensionless form of the ion equations is1 ∂ui∂ui ∂χ¼0þ uiþE ∂t∂x ∂x1 ∂δni ∂þ ½ð1 þ δni Þui ¼ 0E ∂t∂x0ð8:141Þð8:142Þ0Here we have set ni ¼ ðn0 þ δni Þ=n0 ¼ 1 þ δni and have dropped the prime. If thesoliton is stationary in a frame moving with velocity V, the perturbations depend onx and t only through the combination ξ ¼ x x0 Vt. Thus∂∂¼∂x ∂ξ∂∂¼ V∂t∂ξand we obtain after linearizationV ∂ui ∂χ¼0þE ∂ξ ∂ξui ¼V ∂δni ∂uiþ¼0E ∂ξ∂ξδni ¼EχVEuiVð8:143Þð8:144ÞFrom this and the condition of quasineutrality for the slow motions, we obtainδne ¼ δni ¼E2χ:V2ð8:145ÞSubstituting for χ Eq. (8.140), where δn is really δne, we find 211V:δne ¼ juj2 2 14Eð8:146ÞUpon inserting this into Eq.
(8.136), we finally have"1 #2∂u 3 ∂ u1 V2þ1þ Δijuj2 u ¼ 0:∂t 2 ∂x28 E2ð8:147Þ8.8 Equations of Nonlinear Plasma Physics319Comparing with Eq. (8.122), we see that this is the nonlinear Schr€odinger equationif Δ can be neglected and1m=Mq¼8 V 2 m=M3p¼2ð8:148ÞFinally, it remains to show that p and q are related to the group dispersion andnonlinear frequency shift as stated in Eq. (8.123). This is true for V2 m/M. Indimensionless units, the Bohm–Gross dispersion relation (4.30) reads020ω ¼ 1 þ δn þ 3k02ð8:149Þwhere k0 ¼ kλD, and we have normalized ω to ωp0 (the value outside the densitycavity).
The group velocity is00vg ¼dω3k0 ¼ω0dk0ð8:150Þso that0dvgdk0¼3ffi3ω0and0p¼1 dvg2 dk0¼32ð8:151ÞFor V2 E2, Eq. (8.146) gives01 0 2δn ¼ u 4so that Eq. (8.144) can be written02021 0 2ω ¼ 1 u þ 3k4Then001 0 22ω dω ¼ du 0δω /dω040d ju jð8:152Þ2ffi18ð8:153Þ3208 Nonlinear EffectsFrom Eq. (8.148), we have, for V2 E2,0qffi1dω¼ 0 28d ju jas previously stated.If the condition V2 E2 is not satisfied, the ion dynamics must be treated morecarefully; one has coupled electron and ion solitons which evolve together in time.This is the situation normally encountered in experiment and has been treatedtheoretically.In summary, a Langmuir-wave soliton is described by Eq.
(8.125), with p ¼ 3/2and q ¼ 1/8 and with ψ(x, t) signifying the low-frequency part of u(x, t), where u, x,and t are all in dimensionless units. Inserting the exp (iω0t) factor and letting x0and θ0 be zero, we can write Eq. (8.125) as follows:uðx; tÞ ¼ 4A1=2" # 2A 1=2V2VA t xsechðx VtÞexp i ω0 þ336ð8:154ÞThe envelope of the soliton propagates with a velocity V, which is so farunspecified. To find it accurately involves simultaneously solving a Korteweg–deVries equation describing the motion of the density cavity, but the underlyingphysics can be explained much more simply. The electron plasma waves have agroup velocity, and V must be near this velocity if the wave energy is to move alongwith the envelope. In dimensionless units, this velocity is, from Eq. (8.150),0V ffi vg ¼03k0ffi 3kω0ð8:155ÞThe term i(V/3)x in the exponent of Eq. (8.154) is therefore just the ikxfactor indicating propagation of the waves inside the envelope.
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