## Thursday, July 8, 2021

### Solutions To Suggested Plasma Physics Problems

1)  Plot a graph of  F   vs.  v  for the case c s   -  v o   <   0 and indicate the position of v max  on the graph.

Soln.

2)  Integrate  the KdV equation:

a  dV/ dx2   -    (v o     c s ) v  -   v 2/ 2   =  0

And show how the soliton solution:

v   =   3 (v o     c s ) sech 2  [(v o     c s  / 4 a) ½     x' ]

Is obtained.  Given this is in the fluid frame, show what it would be in the lab frame.

Soln.

Multiply the KdV eqn. by v'  and then integrate to obtain:

a/ 2  (v' 2)  =  (v o     c s 2/ 2  -    v3/ 6

We choose the constant of integration to be zero because we want v' = 0  when  v = 0.   Then:

dv/ dx'   =   v'  =   ( 2/ a) ½ [ (v o     c s 2/ 2  -    v3/ 6 ] ½

Or:

dv/  [(v o     c s 2/ 2  -    v3/ 6 ] ½    =   ( 2/ a) ½  dx'

Now integrate both sides to obtain:

ò dv/ Ö (v  2  -  b v  2 )  =   ò ( 2/ a) ½ dx'

Where   b   = 1/  [3 (v o     c s )]

Now let u  =  Ö (1    -   b v)   so  v  = (1 -  u  ) /  b

And du =    (-  b   dv/ 2)/  Ö (1    -   b v)

Then we are in position, with some labor, to work out:

( 2/ a) ½ x'   =  (2/ v o     c s ) ½  ln  (1 - u/  1 + u)

On introducing another compressed factor with an exponential, viz.

exp (g  x')   =    (1 - u/  1 + u)

with  1  - u   =    (1 + u) exp (g  x')

It is straightforward to get to the final soln.   The key penultimate step is:

v  =   1/ b   =   4 [e  x' /2 g  -  e - x' /2 g ]  =   1/ b  sech 2 (g  x'/  2)