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of electrostatic potential in the form

à !
2 (1)
3 (1)
2 (1)
  (1)  (1) (1)  (1)   1     
+ +  +  + + =0 (4.11)
  2   3 2 2 Θ 2 2  2
The coefficients of nonlinear term, A and the dispersion term, B term are

given by

µ ¶ µ 2 2 ¶
1 1  3
2 
 =(4) {  0 2 − +    0 2 − 4 (4.11.a)
  2  
  
⎡ 2 ³ 23 2 ´ ⎤
 2    0 +3
+  0 ⎣ + ³ ´ ⎦ }
  2  23 −  2 3
0
and

1
 =    (4.11.b)
3
8      0  − 23
8      0 +   0
 3  − 23 3
2
0
Equation (4.11) is a 3D-CKP equation which governs the evolution

of the first-order approximation of electrostatic potential corresponding to
dust acoustic waves in a complex plasma. It is important to mention here

that if we neglect the Θ and  dependence, the 3D-CKP equation (4.11),

is reduced to the well known one dimensional K-dV equation.

4.2.1 Solution of 3D-CKP Equation

Using the generalized expansion method, [?] an exact solution for equa-

tion (4.11) is obtained as

h  i
 =  (1) =   2  (4.12)
0
∆

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