Wednesday, June 30, 2021

More Plasma Physics: Looking At Solitons And Soliton Solutions (Of The Korteweg deVries Equation)


In space plasma physics many situations arise in which one will wish to examine the behavior of a solitary wave - say in a particular plasma environment. One such is for nonlinear ion-acoustic waves. 

 The solitary wave (soliton) is just a traveling wave of form:  

u(x, t) =  f(x - ct)  

for some smooth function f that decays rapidly at infinity and under certain conditions can travel without change of shape (see graphic). 

  Consider the partial differential equation:

 utt   - uxx  =   u ( 2 u2  -  1)

Which has a family of solitary wave solutions:

u(x,t) = sech (x cos q +   t sinh q)

We do not expect, for mathematical reasons,  the sum of two such solutions (superposition) to also be a solution.  There is, however, a class of soliton equations which does exhibit a form of nonlinear superposition.  An n-soliton solution then is a solution that is asymptotic to a nontrivial sum of n solitary waves, e.g.

å nt=1 f i  (x - ci  t) as t -> - oo  and to the sum of the same waves:

å nt=1 f i  (x - ci  t  + ri)

with non-zero phase shifts ri  as t -> oo.  I.e. after nonlinear interaction the individual solitary waves pass through each other, keeping their velocities and shapes - but with phase shifts. 

 The  Korteweg de Vries (KdV) equation:

(- v o  + c s   + v)   v / x’  -  m  2 v / x2  

+  a  3 v / x3 =  0

is a well known example of  such a soliton equation admitting nonlinear superposition.  Here c s is the ion sound speed, and v o the  electron thermal speed.  In the form shown we note the appearance of the dissipative term (m  2 v / x2 and that for a soliton to evolve into a shock a dissipative mechanism is needed.  In the more common situation steepening of the wave balances dispersion and one will obtain a wave  form such a shown in the top graphic. We focus more on this case now by adjusting the KdV equation.

  On integrating once,  and excluding the shock evolution term (in m) ,  we obtain:

a  d2 V/ dx2   -    (v o     c s ) v  -   v 2/ 2   =  0 

Which has the same mathematical form as Newton's 2nd law of motion, e.g.  m x" =  F(x)  =   -   x V(x)  

Where V(x) is the potential energy. With some further manipulation we find:

dV/ dx2   =   -   v  V' (x)  =

-  ¶ v [(c s   -  v o) v 2/ 2   +     v3/ 6 ]

For a particle 0f mass a   moving under a potential field given by the quantity in brackets.  We call this quantity the pseudo potential and designate it:

 F (v) =  [(c s   -  v o) v 2/ 2   +     v3/ 6 ]

Which is also known as the Sagdeev potential.  It is left to the industrious reader to do a simple plot of  F (v) vs.  v,   with   v max   shown on v -axis.

For the criteria on  F  to obtain a soliton solution we have:

i)    F / ¶ V ] v= 0  =  0;   2 F¶ v2   <   0

ii)  F   <   0,   For   0   <   v   <   v max

iii)  d F / d V ] v= v max  >  0

We  note here that two graphs of  F   vs.  v are possible,  one for  c s   -  v o   >   0,  the other for  c s   -  v o   <   0.  Since we demand only a localized wave form then it will always be the latter form used, i.e. in further analyses.  One such is to obtain a soliton solution for the KdV equation:

a  d2 V/ dx2   -    (v o     c s ) v  -   v 2/ 2   =  0 

This may be solved exactly i.e. with c s   -  v o   <   0.   The procedure is then to multiply the KdV by v'  and then integrate to obtain:

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

And we choose the constant of integration to be zero because we want v' = 0  when  v = 0.   Working through the process the final solution is found to be:

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

Suggested 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.

2)  Integrate  the KdV equation:

a  d2 V/ 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.

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