## Wednesday, June 22, 2022

### Looking At Electrostatic Shocks (2)- The Shock Profile

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

2j/ d x2  =

4 p e n o  (1/ Ö (1+ 2e j/me v eo  –  1/ Ö 1 – 2 e j)/ mi vio )

=     v (j) / j

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

v (j) =

p n o { me v eo 2 (Ö (1+ 2e j/me v eo )  +  mi vio Ö 1 – 2 e j)/ mi vio )

And:  me v eo 2 =   mi vio   = 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/ Te

And:   x =  x/ le 2    where    le  = v eo  we Ö(2)

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

Which is the dimensionless Poisson equation

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

One requires for the solution:

1)  y  (0)  =  0       Þ    C 1 =  1 +  n ot  +  2/3   F max n or

2) y  (F max ) =  y II (F max  )  Þ    C 1 =  C2

3) y I " (0)  <  0   Þ     2   > n ot  / 1 +  n or /2F max

4) y (F max ) =    0

Þ

1 - exp (F) +   n ot  2 (1 -  Ö(1 – (2 e F)/ M 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: