*must*go read this little book:

- Alex Kasman,
*Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs*(AMS Student Mathematical Library, Volume 34).

[Search for it on Library Genesis, but support the author if at all possible; given the amazing book he produced, he really deserves it.]

I will now proceed to just unapologetically fawn over this goddamned masterpiece until it gets relatively awkward, uhhh, continue doing that for a bit longer, and then finally quickly talk about what this crazy story involves.

### The book

It's been forever since I had

*so*much trouble putting down a math book, or for that matter any kind of technical book. Soliton theory is, today, a very sophisticated branch of mathematical physics (belonging to the theory of integrable systems) that draws heavily on all sorts of high-powered machinery, mostly from algebraic geometry: moduli spaces, theta functions, loop groups, and so on.
Despite all this insane stuff lurking just beneath the surface, the book somehow manages to read like a

*novel*. Like many books in the AMS STML series, the tone is very conversational, and an undergrad could read this fresh out of a basic calculus/linear algebra sequence and understand just about all of it. Kasman's exposition is a cathartic and utterly masterful*tour de force*. Highlights include:- plenty of nice diagrams to help you see what's going on,
- Mathematica code you can run to play around with this stuff to your heart's content, and
- historical interludes that seem to be injected
*extremely*judiciously so as to give the reader's mind the occasional break from the mathematical development.

To top it all off, the book includes pointers to research monographs and papers on these topics, making it very easy to go as far as you like down the rabbit hole (which, here, really is endless).

### Solitons

There are so many worlds colliding here that frankly it's a little hard to even decide where to start, but the concept of a soliton is probably as good a place as any. Long story short, back in the 19th century, some guy John Scott Russell was watching a boat getting towed down a very narrow canal by a pair of horses, and noticed that when the boat suddenly stopped, a very well-defined hump of water formed, which he followed on horseback as it propagated, seemingly undisturbed, down the canal. He found this fascinating; it seems most people at the time responded along the lines "cool, but like, whatever".

About 50 years later, in 1895, the Korteweg–de Vries ("KdV") paper appeared, where they came up with the following equation as a model for shallow water waves: \[ \frac{\partial u}{\partial t} - 6u \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3} = 0. \] Okay, it wasn't

About 50 years later, in 1895, the Korteweg–de Vries ("KdV") paper appeared, where they came up with the following equation as a model for shallow water waves: \[ \frac{\partial u}{\partial t} - 6u \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3} = 0. \] Okay, it wasn't

*exactly*like that in their paper; they had a bunch of physical constants in there, but that's what it boils down to mathematically. It's called the**KdV equation**. When I first saw this I was immediately annoyed by the seemingly random "6", but it turns out it's only there for conventional reasons; in fact by applying simple transformations you can get whatever coefficients you want to appear in front of the three terms. Anyway, the upshot is that once you fix an "initial profile" \( u_0(x) = u(x,0) \), solving the KdV equation tells you what happens as time passes, i.e. it tells you \( u(x,t) \).
The KdV equation is a partial differential equation (PDE). Using the subscript notation for partial derivatives we can rewrite it as \( u_t - 6uu_x + u_{xxx} = 0 \). Due to the second term, which involves \( uu_x \), it is

*nonlinear*. As a consequence, the superposition \( u+v \) of two solutions \( u \) and \( v \) to this equation will in general*not*be another solution. On the other hand, due to the third term \( u_{xxx} \), it is*dispersive*. To understand this, suppose for a moment that there was no nonlinear term. Then if you have some nice, localized "hump", and then you watch how it evolves, then typically this serendipitous equilibrium would quickly be destroyed, and your "hump" will fall apart into a horrendous mess: dispersion. This is the miracle of KdV: due to the presence of*both*the nonlinear and dispersive terms, they "compete" with each other in some sense, and thus allow the possibility for solitary waves that simply propagate without changing shape (and even pass through each other virtually unaffected)! Hence, the term**soliton**was coined.
Cool! So how do we solve it? Well, that's the thing, people found one relatively simple solution, and then pretty much had no idea how to proceed. Even though there is only one spatial dimension, it's a

*nonlinear*PDE, and solving those typically ranges from "sort of tricky", to "Japanese hard mode", to "E8 ROOT SYSTEM SCRAWLED IN CHALK ON FLOOR BURNING SPRINGER BOOKS ON ALTAR WHILE HISSING IN TONGUES".
So, for over 50 years, the KdV equation laid more or less dormant, and nearly even forgotten. People moved on with their lives. However, as we will see, the story didn't end there. Things were destined to take an absolutely unexpected turn, leading us on a wild chase through a mysterious jungle, to a singular insight that would revolutionize the area forever.

### Integrability

Integrability is a property that certain dynamical systems possess. Very roughly, it means they evolve in a "nice", rather than a chaotic, way. Their solutions can also be explicitly written down, for varying degrees of "explicit". For Hamiltonian systems, there is a single, all-encompassing definition of integrability, and the theory here is very well-developed. Passing from here to the setting of PDEs is basically passing from classical mechanics (e.g. studying the motion of some finite number of particles), where we have only finitely many degrees of freedom, to field theory (e.g. studying the motion of a vibrating string), where we have

*infinitely many*degrees of freedom. This complicates matters considerably: so far there seems to be no "perfect" definition of integrability for PDEs. We do have some ideas, though. Three in particular stand out:- the existence of a so-called
*Lax representation*, which roughly speaking is a way of expressing the PDE in question as the compatibility condition of an overdetermined linear system; - the so-called
*Painlevé property*, a certain condition on how much "dependence" is exhibited by the singularities of a differential equation; and - perhaps most promisingly,
*expressibility as a dimensional reduction of the anti-self-dual Yang–Mills equations*(almost all famous examples in 2 and 3 dimensions fall under this umbrella).

### The GGKM paper: the rise of "inverse scattering"

In 1967, there appeared the paper titled

*Method for Solving the Korteweg*–*deVries Equation*by Gardner, Greene, Kruskal and Miura, describing at last an ingenious method of solving the KdV equation. Surprisingly enough, it turns out that if you take a solution of the KdV equation and interpret it as a*potential*of a Schrödinger operator (wait, whaaa?!), then all kinds of magic happens. As you let the potential evolve according to KdV, the spectrum of the corresponding Schrödinger operator*does not change*!
Now, if you know the "scattering data" of the Schrödinger operator at a given time (which has to do with how an incoming disturbance is reflected off, or transmitted through, the potential), it turns out that you can in fact recover the potential at that time (thus we say you are "solving the inverse scattering problem"; this is essentially how physicists draw conclusions about elementary particles from particle accelerator data). Then, since the scattering data turns out to evolve in a nice way that we can understand, by solving an integral equation, we can recover the

*evolution of the potential*, in other words, we can solve the KdV equation for the given initial data.
Like most good ideas, this

**inverse scattering method**turned out to be much broader in scope than initially expected. In fact it can be used to attack a wide array of other nonlinear PDEs, such as the Kadomtsev–Petviashvili equation (a generalization of KdV to two spatial dimensions), the nonlinear Schrödinger equation, the Ernst equation (equivalent to the vacuum Einstein field equations for a stationary, axisymmetric spacetime), and the sine-Gordon equation. These equations have been derived*over*and*over*from a host of natural problems – some coming from physics, some coming from pure geometry. The key, really, is the Lax representation: in the case of KdV, it's what provides the link to the Schrödinger equation (which is*linear*!).
In fact, if you read Kasman's book, you will see there is a sense in which the KP equation falls right out of the Plücker relations that cut out the Grassmann cone. Let that sink in for a moment: mathematical physics on the one hand (KP is a nonlinear PDE modelling WATER WAVES!),

*pure algebraic geometry*on the other. One just can't*help*but marvel at this! In fact, there is a whole KP*hierarchy*which arises just as naturally; see Segal–Wilson.### Schottky problem

A Riemann surface is a one-dimensional complex manifold: a geometric object that locally looks like a patch of the complex plane. These surfaces historically arose from scissors-and-glue constructions (pasting several complex planes together along "branch cuts") that were used to understand the behaviour of multi-valued functions of a complex variable, such as the square root "function" \( z \mapsto \sqrt{z} \), which is really two-valued, or the logarithm, which is actually infinite-valued (and therefore properly defined on a kind of "infinite parking lot"). Every compact Riemann surface \( \Sigma \) is

It turns out that the holomorphic differentials (1-forms) on \( \Sigma \) form a vector space of dimension precisely \( g \). We can choose a basis \( \{ \omega_1, \ldots, \omega_g \} \) for this space which is "adapted" to the system of curves above in the sense that the integrals \( \oint_{a_k} \omega_j = \delta_{jk} \). Then the "interesting information" is contained in the integrals \( \pi_{jk} := \oint_{b_k} \omega_j \). Thus we have extracted a matrix \( \Pi = (\pi_{jk}) \) of complex numbers called the

There is a problem in algebraic geometry that asks for a characterization of Jacobian varieties among all abelian varieties. Stated differently: can we characterize the locus of

*topologically*either a sphere, or just a \( g \)-holed "donut" for \( g \geq 1 \). We can then choose a family of curves \( a_1, \ldots, a_g, b_1, \ldots, b_g \) on the surface, that look like this:It turns out that the holomorphic differentials (1-forms) on \( \Sigma \) form a vector space of dimension precisely \( g \). We can choose a basis \( \{ \omega_1, \ldots, \omega_g \} \) for this space which is "adapted" to the system of curves above in the sense that the integrals \( \oint_{a_k} \omega_j = \delta_{jk} \). Then the "interesting information" is contained in the integrals \( \pi_{jk} := \oint_{b_k} \omega_j \). Thus we have extracted a matrix \( \Pi = (\pi_{jk}) \) of complex numbers called the

**period matrix**(there is a sense in which the choices we made above didn't really matter), and we can show this matrix is symmetric, and has positive-definite imaginary part. The set of all \( g \times g \) complex matrices with these two properties is called the**Siegel upper half-space**\( \mathfrak{S}_g \)**of genus**\( g \). For the number theorists in the audience, incidentally this is the object on which the fascinating*Siegel modular forms*live. Anyway, one then uses the columns of \( \Pi \) to form a lattice \( \Lambda \) in \( \mathbf{C}^g \cong \mathbf{R}^{2g} \), and then goes on to define the Jacobian \( J(\Sigma) := \mathbf{C}^g / \Lambda \) of the curve, determines conditions under which meromorphic functions exist on the surface with prescribed behaviours, and so on. This stuff is all very classical and was worked out in the 19th and 20th centuries. It beautifully encapsulates all the (previously somewhat scattered) knowledge about elliptic integrals (and more generally, abelian integrals) into an abstract, geometric framework.There is a problem in algebraic geometry that asks for a characterization of Jacobian varieties among all abelian varieties. Stated differently: can we characterize the locus of

*period matrices of Riemann surfaces*in the Siegel upper half-space \( \mathfrak{S}_g \)?
This question is a very classical one, and has been studied extensively. To this day, there is a sense in which it still hasn't been solved in a

...

Alright, I'm tired. There are many more ingredients to discuss, even at this fundamental level, such as \( \tau \)-functions, and the Wronskian determinant that allows us to "combine" solutions despite the nonlinearity (!), and so on, but that's enough for today.

Go read Kasman's book, and then once you're starving for more, you can move on for example to Hitchin–Segal–Ward's book

*completely*satisfactory way, but the answer we have so far is already rather striking: to formulate it, note that for any \( \tau \) in \( \mathfrak{S}_g \), we can define a function of several complex variables called its Riemann theta-function \( \Theta_\tau : \mathbf{C}^g \to \mathbf{C} \): \[ \Theta_\tau(\mathbf{z}) := \sum_{\mathbf{m} \in \mathbf{Z}^g} \exp \left( 2\pi i \left( \frac{1}{2} \mathbf{m}^\top \tau \mathbf{m} + \mathbf{m}^\top \mathbf{z} \right) \right). \]Then the period locus consists precisely of those matrices \( \tau \) in \( \mathfrak{S}_g \) whose corresponding Riemann theta-function \( \Theta_\tau \) satisfies the KP equation!...

Alright, I'm tired. There are many more ingredients to discuss, even at this fundamental level, such as \( \tau \)-functions, and the Wronskian determinant that allows us to "combine" solutions despite the nonlinearity (!), and so on, but that's enough for today.

Go read Kasman's book, and then once you're starving for more, you can move on for example to Hitchin–Segal–Ward's book

*Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces*.