Looking At Electrostatic Shocks (2)- The Shock Profile

The Poisson equation in the context of electrostatic shocks can be written:

d ²j/ d x²  = 

 4 p e n _o  (1/ Ö (1+ 2e j/m_e v _eo ²  –  1/ Ö 1 – 2 e j)/ m_i v_io ² )

 =    ¶ v (j) / ¶ j 

Where v (j) is a pseudo-potential or the Sagdeev potential.  On integrating:

v (j) = 

- 4 p n _o { m_e v _eo ² (Ö (1+ 2e j/m_e v _eo ² )  +  m_i v_io ² Ö 1 – 2 e j)/ m_i v_io ² )

And:  m_e v _eo ² =   m_i v_io ²   = const.

The wave speed is fixed with respect to the relative motion of the electrons, ions.  This allows a determination of the wave speed, i.e. we look for a class of nonlinear waves which satisfy this equality. (This leads to the "BGK" waves.)

In pursuing shock solutions-solitons it is customary to make the above non-dimensional to simplify it. To that end, one can use the non-dimensional constant:

 F = e j/ T_e

 And:   x =  x/ l_e ²    where    l_e  = v _eo  / w_e Ö(2)

Then:   d ² F/ d x ²  =   exp (F) -  1Ö(1 – (2 e F)/ (M _s )²  =    - ¶  y/ ¶ j 

Which is the dimensionless Poisson equation

Where:   Y  =  exp (F) - 1/ Ö(1 – (2 e F)/ M _s ²)  +  C

 One requires for the solution:

1)  y _I  (0)  =  0       Þ    C 1 =  1 +  n _ot  M _s ² +  2/3   F _max n _or 

2) y _I  (F _max ) =  y _II (F _max  )  Þ    C 1 =  C2

 3) y _{I "} (0)  <  0   Þ     M _s ²   > n _ot  / 1 +  n _or /2F _max  

_{4) y (F max ) =    0}  

Þ 

 1 - exp (F) +   n _ot  M _s ² (1 -  Ö(1 – (2 e F)/ M _s ²)  +  2/3   F _max n _or = 0

(Also:   n _ot    +   n _or   =   1)

5) y _II (F)   >   0

I.e. One solves for y _II (F)  =   0  over:

F _min   <  F  <   F _max      And seeks a monotonic function

For reference, a sketch of the graph of the relevant potentials is given below which in terms of the dimensionless Poisson solutions can be regarded as the shock profile: