5Symmetry methods in PDEs

II Integrable Systems



5 Symmetry methods in PDEs
Finally, we are now going to learn how we can exploit symmetries to solve
differential equations. A lot of the things we do will be done for ordinary
differential equations, but they all work equally well for partial differential
equations.
To talk about symmetries, we will have to use the language of groups. But
this time, since differential equations are continuous objects, we will not be
content with just groups. We will talk about smooth groups, or Lie groups. With
Lie groups, we can talk about continuous families of symmetries, as opposed to
the more “discrete” symmetries like the symmetries of a triangle.
At this point, the more applied students might be scared and want to run
away from the word “group”. However, understanding “pure” mathematics is
often very useful when doing applied things, as a lot of the structures we see in
the physical world can be explained by concepts coming from pure mathematics.
To demonstrate this, we offer the following cautionary tale, which may or may
not be entirely made up.
Back in the 60’s, Gell-Mann was trying to understand the many different
seemingly-fundamental particles occurring nature. He decided one day that he
should plot out the particles according to certain quantum numbers known as
isospin and hypercharge. The resulting diagram looked like this:
So this is a nice picture, as it obviously formed some sort of lattice. However, it
is not clear how one can generalize this for more particles, or where this pattern
came from.
Now a pure mathematician happened to got lost, and was somehow walked
into in the physics department and saw that picture. He asked “so you are also
interested in the eight-dimensional adjoint representations of
su
(3)?”, and the
physicist was like, “no. . . ?”.
It turns out the weight diagram (whatever that might be) of the eight-
dimensional adjoint representation of
su
(3) (whatever that might be), looked
exactly like that. Indeed, it turns out there is a good correspondence between
representations of
su
(3) and quantum numbers of particles, and then the way to
understand and generalize this phenomenon became obvious.

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