Solitary electron plasma waves

Solitary electron plasma waves

Volume 59A, number 5 PHYSICS LETTERS 27 December 1976 SOLITARY ELECTRON PLASMA WAVES M.Y. YU Institut für Theoretische Physik, Ruhr Universität, 46...

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Volume 59A, number 5


27 December 1976

SOLITARY ELECTRON PLASMA WAVES M.Y. YU Institut für Theoretische Physik, Ruhr Universität, 463 Bochum, F.R. Germany Received 23 September 1976 We show that solitons can appear in an electron plasma. These solitons correspond to localized density depressions which travel at sub-thermal speeds. Analytic expressions are obtained for small amplitude near-thermal speed solitons.

Although ion acoustic solitons [1, 2,3] have been known for some time, solitons associated with nonlinear plasma waves seem to have never been discussed previously. In this Letter, we show by means of both the isothermal and the adiabatic electron gas models that steady state moving electron plasma solitons exist. The equations governing an electron gas are

a~n+a~(nu)=o, (~+v~~)v= (elm) ~ PnT

(1/nm) axP, ,

1.+ va~)P/n~ = 0 adiabatic,

a 2 Ø=—4ire(n0—n), X


where the temperature T is constant and the ions are assumed to form a uniform a uniform background. The notation is standard. Going over onto a frame moving with speed V, and assuming steady state in this frame, we can integrate the resulting equations. One obtains after eliminating F, nv’—M, (5) lnn isothermal,


2 =




(6b) 17\

~‘ where x, v, n and 0 have been nondimensionalized by X = (T/47rn 2)1/2, 0T (Tim)1!2, n 0e 0 and Tie respectively. The Mach number M is defined by V/VT. In the adiabatic model, we have defined T= 3P0/2n0, where p0 and n0 correspond to the undisturbed conditions at infinity. Eqs. (5) to (7) can be combined to yield x


2 2 M a~22

2 +




inn isothermal n2 adiabatic

(8 a) (8b)






n2 ln n


n3 ÷ n2 +M2n 2/2)

(M2—n2)2 n4(M2 +n4 +2M2n

(3a) (3b)


Integrating (8a) and (8b) once, we obtain respectively ~ n~2=










2n5/3 + M2n2 n4)2

(9a) —



where the conditions n(°°) = 1 and a~n(o0) 0 have been used to determine the constants of integration. Eq. (9a) and (9b) are in the form of the energy integral of a classical particle, whose effective potential energy is given by ‘J!(n). We can therefore analyze them to see if localized solutions exist for n [1,3]. First,we note that for both models ‘I’(l) = a~’I’(l)= 0, and a~’T.’(l)<0 if M < 1. Secondly, there exists n = nM
(lOb) such that ‘~‘(~m) = 0 and Wn(nm)
Volume 59A, numberS



0 = 0 for 0<

2X otherwise.



2)h/2[l+sech~(2/3)1/21 (2(1 M 2~2(l M2)~2131/2 + 2 sech 1(2/3)1/21’

8 .7


for solitons in an isothermal and an adiabatic electron gas respectively. The shape of these solitons can easily be visualized to be similar to that of a simple sech2~ soliton 121, except for the steep drops at the shoulders. Although fig. I indicates that the solitons can have speeds approaching zero, it is expected that for M < (~~~/M)1/2,where M is the ion mass, ion motion cannot be neglected. When the latter effect is included, one recovers the ion acoustic soliton of Sagdeev Ill. Clearly, no such transition can occur between small amplitude electron plasma and ion acoustic solitons. We have not investigated the stability of the solitons

~ /



.3 .2














found here. For this purpose, extensive numerical work is required.

Fig. 1. The density at maximum soliton amplitude versus the Mach number.

Finally, we mention that by choosing different boundary conditions for the potential ‘If(n), one can

(1 -- M2)~,i2+ 4~n3/3


(11 a)



M2)~n2+ 8~n3/3 4(1~M2+46n)2 .


These equations can also be derived from the set (1.) to (4) by assuming an appropriate ordering scheme and some algebra. Upon integration of equations (11), we obtain the following transcendental representations


M2) sech2



[lxi on

27 December 1976


(12 a) 1/2 (i

X 0 +


3(1-_M2)h2 8

M2 + 4~)1/211




1i~ ~2(1

M2)1/2 (l2b)

rIxI—x 0

21/2 XlL—-~+3



2 1 —M


also construct cnoidal wave structures for the nonlinear plasma waves. This work is supported by the Sonderforschungsbereieh 162 Plasmaphysik Bochum/JUlich.

References [1] R.Z. Sagdeev, in: Reviews of Plasma Physics, vol. 4 (Consultants Bureau, New York, 1966) p. 23. 121 II. Washimi and T. Taniuti, Phys. Rev. Lett. 17 (1966) 996. [31 I.E. Chen, Introduction to plasma physics (Plenum, New York, 1974) p. 249.