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model is given by continuity equation [10]

 (    )
 
+ =0 (1.6)
 

and momentum equation


   
+   = −  (1.7)
  

The system is closed by Poisson’s equation

2
 
=  0 −    (1.8)
 2

where    and  0 are the electron velocity, electric potential and the

unperturbed ion density, respectively.
More than 80 % of the observed plasma state can be investigated by

fluid model. Another approach using lumped macroscopic variables and

hydrodynamic conservation equations is called the magnetohydrodynamic
(MHD) theory [3, 4].

1.4 Perturbation Analysis for Nonlinear Waves

The equations describing plasmas are coupled nonlinear partial differ-

ential equations and can not be solved in exact form. Thus, system in-
formation can be obtained by perturbation approximations and numerical

solutions. Perturbation techniques are analytical methods for deriving

approximate solutions describing nonlinear propagation and wave inter-
actions. Accordingly, reductive perturbation approximation is related to

plasma theory. The basic concept of this method was specified by Gardner-

8