Integrability of Nonlinear Systems

and upon comparing (3.23)–(3.24) we see that
u(x) =− ∂ 1
∂x π
Nonlinear Waves, Solitons, and IST 15
∫ ∞
−∞
r(ζ)N(x, ζ)dζ. (3.25)
Thus, given the scattering data, r(ζ), we can construct a solution of the integral
equation (3.22) and the potential u(x).
Equations (3.22), (3.25) can be simplified by looking for solutions with a
certain structure,
∫ ∞
}
N(x, k) =e
{1+
2ikx K(x, s)e ik(s−x) ds . (3.26)
Substituting (3.26) into (3.22) and operating on the result with
1
2π
∫ ∞
−∞
x
dk e ik(x−y)
for y>x, we find the so-called Gel’fand-Levitan-Marchenko equation (GLM),
K(x, y)+F (x + y)+
∫ ∞
x
∫ ∞
K(x, s)F (s + y) ds =0, (y >x), (3.27)
where F (x) =F c (x) = 1 r(k)e ikx dk. Similarly, substitution of (3.26) into
2π −∞
(3.25) yields
u(x) =2 ∂ K(x, x). (3.28)
∂x
So far we have not allowed for the possibility that a(k) can vanish for Im k>
0. If a(k) vanishes at k j = iκ j , κ j > 0, j =1,...N, the final result is that the
GLM equation is only modified by changing the function F (x):
F (x) =F c (x)+F d (x) (3.29)
where F c (x) is given below (3.27) and F d (x) is defined by
F d (x) =
N∑
C j exp(−κ j x), (3.30)
j=1
where C j are certain normalizing coefficients related to φ(x, k j )=Ĉjψ(x, k j ),
where C j = −iĈj/a ′ (k j ).
Thus the complete solution of the inverse problem is as follows. Given the
scattering data S(k) ={r(k), {κ j ,C j } N j=1 }, we form F (x) from (3.29), solve the
GLM equation (which results from the RHBVP (3.11) or (3.17)) for K(x, y)
and obtain the potential u(x) from (3.28). In fact, the procedure for inverse
scattering of the 2×2 problem (2.5) and many other scattering problems related