Higher-order nonlinear contributions to ion-acoustic waves in a plasma consisting of adiabatic warm ions

A. Esfandyari-Kalejahi¤, I. Kourakis† and P. K. Shukla †
¤Azarbaijan University of Tarbiat Moallem, Faculty of Science, Department of physics, 51745-406, Tabriz, Iran.
†Ruhr-Universität Bochum, Institut fürTheoretische Physik IV, D-44780 Bochum, Germany.
Abstract. Ion-acoustic solitary waves in a collisionless plasma consisting of adiabatic warm ions, a weakly relativistic
electron beam and non-isothermal electrons, are studied by using the reductive perturbation method. The basic set of model
equations is reduced to a modified Korteweg-de Vries equation (mKdV) for the first order electric potential correction, and to
a linear inhomogeneous equation for its second-order counterpart. It is remarked that the effect of higher order nonlinearity
results in an increased soliton amplitude for one of these modes, in addition to deforming the soliton’s shape (from a simple
positive pulse to a W-shaped excitation), while in the case of the three other modes, the effect of higher-order nonlinearity is
to increase the soliton’s amplitude without deforming its shape (a simple pulse is obtained). The effects of the electron beam
density and the ion temperature on the existence and propagation of solitary waves are also briefly studied.
Keywords: Ion-acoustic waves, Relativistic electron beam, Two fluid plasma, Modified Korteweg-de Vries equation, Higher-order nonlinearity
PACS: 52.35.Mw,52.40.Mj,52.30.Ex
THE MODEL EQUATIONS
We assume a collisionless plasma consisting of warm ions, non-isothermal electrons and weak (for the sake of
simplicity) relativistic electron beam. We shall consider the lowest-order ion-acoustic solitary waves, anticipating
the effect of higher-order nonlinearity on them. A reductive perturbation method will be employed. Wave propagation
parallel to the external magnetic.

NUMERICAL ANALYSIS AND CONCLUSION
The dispersion relation (8) is quartic in S, and thus shows that the inclusion of a finite ion temperature, nonisothermal
electrons and relativistic electron beam gives rise to four ion-acoustic modes propagating with different
phase velocities S1, S2, S3 and S4. For instance, the dispersion relation has only two real roots S1, S2 for v0 = 1:8,
c = 30, m = 1=1836, l = 0:5, b = ¡0:99; a = 0:0001 and s = 0:2¡0:3. Figures (1)-(2) show curves for the lowestorder
modified KdV soliton (8) and for the second-order one soliton solution (19). It is remarked that the effect of
higher order nonlinearity results in an increased soliton amplitude for one of these modes S1, while deforming the
soliton’s shape (from a simple positive pulse to a W-shaped excitation), whilst in the case of the second mode S2, the
effect of higher-order nonlinearity is to increase the soliton’s amplitude without deforming its shape (a simple pulse
is obtained). In addition, increasing the value of s increases (decreases) the soliton’s amplitude in the case of mode
S1 (S2), for fixed a. Generally, the analysis reveals that four distinct ion-acoustic modes, which propagate at different
phase velocities, may occur in this plasma system. Two of these modes exist for all values of the electron beam to
background electron density ratio a and ion to free electron temperature ratio s . The amplitude of (mKdV) solitons
decreases (increases) as a (s , respectively) increases, while the inverse behavior is witnessed by the soliton width,
for these two modes. On the other hand, the two remaining modes exist only for small values of a and high values of
s ; in fact, the phase speed of these modes becomes complex at some range of values of a and s . The amplitude and
width associated to these modes decreases (increases) as a( s , respectively) increases. Finally, it is remarked that the
effect of higher order nonlinearity results in an increased soliton amplitude for one of these modes, while deforming
the soliton’s shape (from a simple positive pulse to a W-shaped excitation), while in the case of the (three) remaining
modes, the effect of higher-order.

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