3Inverse scattering transform
II Integrable Systems
3 Inverse scattering transform
Recall that in IB Methods, we decided we can use Fourier transforms to solve
PDE’s. For example, if we wanted to solve the Klein–Gordon equation
u
tt
− u
xx
= u,
then we simply had to take the Fourier transform with respect to x to get
ˆu
tt
+ k
2
ˆu = ˆu.
This then becomes a very easy ODE in t:
ˆu
tt
= (1 − k
2
)ˆu,
which we can solve. After solving for this, we can take the inverse Fourier
transform to get u.
The inverse scattering transform will follow a similar procedure, except it is
much more involved and magical. Again, given a differential equation in
u
(
x, t
),
for each fixed time
t
, we can transform the solution
u
(
x, t
) to something known
as the scattering data of
u
. Then the differential equation will tell us how the
scattering data should evolve. After we solved for the scattering data at all
times, we invert the transformation and recover the solution u.
We will find that each step of that process will be linear, i.e. easy, and this
will magically allow us to solve non-linear equations.