Obtaining Soliton Solutions For The Korteweg deVries Equation

 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 a soliton equation admitting nonlinear superposition, see e.g. a graphical representation of a soliton solution here:

In the KdV equation c _s is the ion sound speed, and v _o the  electron thermal speed.  In the form shown, note the appearance of the dissipative term:

 (m ¶ ² v / ¶x²)  

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 we obtain a wave form such as shown in the graphic

  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  x"  =  d² x/dt ²

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

a d² V/ 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 ) v 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 Problem:

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.