Other approaches to the formulation of the inverse scattering problem, with particular reference to the KdV equation, may be found in Lamb (1980), Ablowitz & Segur (1981), Calogero & Degasperis (1982), Dodd, Eilbeck, Gibbon & Morris (1982).
Exercises
Q3.1 Let y,, v2 be two solutions of the differential equation
y" + p(x)y' + q(x)y = 0. Define the Wronskian W, of y\ and y2, and show that
W + p(x)W = 0.
Hence deduce that either W = 0 for all x or W never vanishes (provided
x and p(x) remain finite). Q3.2 Show that a continuous eigenfunction,      of equation (3.1), and any
discrete eigenfunction, i//„, are orthogonal. Q3.3 Find, if they exist, the eigenvalues and eigenfunctions of
V + (a - U0W = 0,
where U0 is any real constant. Q3.4 Two classical scattering problems. Find the eigenvalues and eigenfunctions of
!/>" + {/ u(x)}l/>=0
in these two cases:
(i) u(x)
U0, 0<x<l, 0, x<0, x>l,
where L'„ is any real constant. *(ii) u(x)= -U0S(x)-UlS(x-\l
where L0 and U1 are positive constants, and show that there is only one discrete eigenfunction if (U0 + 1/1)/(l/01/!) > 1. Q3.5 Another scattering problem. Find the eigenvalues and eigenfunctions of
iA" + {A-u(x)}iA = 0,
if u{x) is the step potential
[0, x<0 o, x > 0,
where U0 is a positive constant. Show, in particular, that there is a
continuous eigenfunction, no discrete eigenfunction, and an eigenfunction
which decays as x -► + oo but is oscillatory in x < 0. Q3.6 Relate the scattering problem with the potential u(x) = — U0 sech2 fix, for
some positive constant fi, to the problem discussed in example (ii), §3.2. Q3.7 A reflectionless potential. Find the discrete eigenfunctions for N = 3, where
u(x)= —N(N + 1) sech2x (see example (ii), §3.2).