2Partial Differential Equations
II Integrable Systems
2.1 KdV equation
The KdV equation is given by
u
t
+ u
xxx
− 6uu
x
= 0.
Before we study the KdV equation, we will look at some variations of this where
we drop some terms, and then see how they compare.
Example. Consider the linear PDE
u
t
+ u
xxx
= 0,
where
u
=
u
(
x, t
) is a function on two variables. This admits solutions of the
form
e
ikx−iωt
,
known as plane wave modes. For this to be a solution,
ω
must obey the dispersion
relation
ω = ω(k) = −k
3
.
For any
k
, as long as we pick
ω
this way, we obtain a solution. By writing the
solution as
u(x, t) = exp
ik
x −
ω(k)
k
t
,
we see that plane wave modes travel at speed
ω
k
= −k
2
.
It is very important that the speed depends on
k
. Different plane wave modes
travel at different speeds. This is going to give rise to what we call dispersion.
A general solution is a superposition of plane wave modes
X
k
a(k)e
ikx−iω(k)t
,
or even an uncountable superposition
Z
k
A(k)e
ikx−iω(k)t
dk.
It is a theorem that for linear PDE’s on convex domains, all solutions are indeed
superpositions of plane wave modes. So this is indeed completely general.
So suppose we have an initial solution that looks like this:
We write this as a superposition of plane wave modes. As we let time pass,
different plane wave modes travel at different speeds, so this becomes a huge
mess! So after some time, it might look like
Intuitively, what gives us the dispersion is the third order derivative
∂
3
x
. If we
had ∂
x
instead, then there will be no dispersion.
Example. Consider the non-linear PDE
u
t
− 6uu
x
= 0.
This looks almost intractable, as non-linear PDE’s are scary, and we don’t know
what to do. However, it turns out that we can solve this for any initial data
u
(
x,
0) =
f
(
x
) via the method of characteristics. Details are left on the second
example sheet, but the solution we get is
u(x, t) = f(ξ),
where ξ is given implicitly by
ξ = x − 6tf (ξ)
We can show that
u
x
becomes, in general, infinite in finite time. Indeed, we have
u
x
= f
0
(ξ)
∂ξ
∂x
.
We differentiate the formula for ξ to obtain
∂ξ
∂x
= 1 − 6tf
0
(ξ)
∂ξ
∂x
So we know
∂ξ
∂x
becomes infinite when 1+6
tf
0
(
ξ
) = 0. In general, this happens in
finite time, and at the time, we will get a straight slope. After that, it becomes
a multi-valued function! So the solution might evolve like this:
This is known as wave-breaking.
We can imagine that −6uu
x
gives us wave breaking.
What happens if we combine both of these effects?
Definition (KdV equation). The KdV equation is given by
u
t
+ u
xxx
− 6uu
x
= 0.
It turns out that this has a perfect balance between dispersion and non-
linearity. This admits very special solutions known as solitons. For example, a
1-solution solution is
u(x, t) = −2χ
2
1
sech
2
χ
1
(x − 4χ
2
1
t)
.
The solutions tries to both topple over and disperse, and it turns out they
actually move like normal waves at a constant speed. If we look at the solution,
then we see that this has a peculiar property that the speed of the wave depends
on the amplitude — the taller you are, the faster you move.
Now what if we started with two of these solitons? If we placed them far
apart, then they should not interact, and they would just individually move to
the right. But note that the speed depends on the amplitude. So if we put a
taller one before a shorter one, they might catch up with each other and then
collide! Indeed, suppose they started off looking like this:
After a while, the tall one starts to catch up:
Note that both of the humbs are moving to the right. It’s just that we had to
move the frame so that everything stays on the page. Soon, they collide into
each other:
and then they start to merge:
What do we expect to happen? The KdV equation is a very complicated non-
linear equation, so we might expect a lot of interactions, and the result to be a
huge mess. But nope. They pass through each other as if nothing has happened:
and then they just walk away
and then they depart.
This is like magic! If we just looked at the equation, there is no way we could
have guessed that these two solitons would interact in such an uneventful manner.
Non-linear PDEs in general are messy. But these are very stable structures in
the system, and they behave more like particles than waves.
At first, this phenomenon was discovered through numerical simulation.
However, later we will see that the KdV equation is integrable, and we can in
fact find explicit expressions for a general N-soliton equation.