Fundamentals of Plasma Physics

15.7 Assignments 457

Drazin and Johns (1989)). This theory is called inverse scattering (Gardner et al. 1967)
and involves mapping the non-linear equation to a special linear equation which is then
solved and then mapped back again to give the time evolution of the nonlinear solution.
The special linear equation describes a quantum mechanical particle trapped in a potential
well whereχ(ξ,t)plays the role of the potential. In particular, the linear equation is of
the form∂^2 ψ/∂ξ^2 +(λ−χ(ξ,t))ψ= 0whereλis an eigenvalue and timetis treated
as a parameter. When this linear equation is solved forχ(ξ,t)and the solution is inserted
in the KdV equation it is found that∂λ/∂t= 0.This invariance of the eigenvalues is a
key feature which makes possible the {nonlinear→linear} and then {linear→nonlinear}
mappings required to construct the solution.

15.7 Assignments

1. Pump depletion for a system of three coupled oscillators.
   (a) SupposeA 1 = 0att= 0butA 2 andA 3 are finite. What is the value of the
   constant in Eq.(15.24)?
   (b) Suppose thatsinθis not zero. WillA 1 become finite at timest >0?IfA 1
   becomes finite, what constraint does Eq.(15.24) put on the value of cosθand
   hence onsinθ?
   (c) Use Eqs.(15.21) to writeA^21 (t)andA^22 (t)in terms ofA^23 (t)and the initial con-
   ditionsA^20 (0),A^21 (0), andA^23 (0).
   (d) Square both sides of Eq.(15.15c) and use the results in (b) (c) above to obtainan
   equation of the form

1

2

(

dA 3

dt

) 2

+U(A 3 )=E (15.192)

whereEis a constant. Sketch the dependence ofU(A 3 )onA 3 indicating the

locations of maxima and minima.

(e) Consider Eq.(15.192) as the energy equation for a pseudo-particle with veloc-

itydA 3 /dtin a potential wellU(A 3 ).What is the total energy of this pseudo-

particle in terms ofA^20 (0),A^21 (0), andA^23 (0)?What does the position of the

pseudo-particle in the potential well correspond to?

(f) Considering the initial conditions given in (a), where is the pseudo-particle ini-

tially located in its potential well? Sketch the qualitative time dependence ofA 3

and indicate how features of this time dependence correspond to the locationof

the pseudo-particle in its potential well.

(g) Give an integral expression for the time required forA 3 to go to its first zero.

(h) Solve Eqs.(15.15a)-(15.15c) numerically for the initial conditions given in (a)

and use the numerical results to show that the time dependence ofA 3 can be

interpreted as the position of a pseudo-particle in a potential well prescribed by

Eq.(15.192). Note: The reduction of the pump wave amplitude as it transfers en-

ergy to the daughter waves is called pump depletion;the mathematical behavior