Sebastian Klein
Universität Mannheim, Germany

Lax, Korteweg-de Vries, Schrödinger, (sinh-)Gordon:
Integrable systems and spectral theory
The 11th GEOSOCK Seminar “Spectral Curves, Integrable Systems and Diﬀerential Geometry”(Osaka University - Osaka City University - Kobe university - Kyushu University
Joint Geometry Seminar).
Abstract. The topic of this pair of lectures is the spectral theory for integrable systems. It
is not easy to give a precise definition of when one calls a dynamical system integrable; in thewords of Hitchin1 , “integrability of a system of diﬀerential equations should manifest itself throughsome generally recognizable features: the existence of many conserved quantities; the presence ofalgebraic geometry; the ability to give explicit solutions.” For the present lectures we are interested
in infinite-dimensional integrable systems defined by a partial diﬀerential equation in two variables.
In the context of integrable systems, “spectral theory” refers to an assortment of methods deriving
conserved quantities for integrable systems by means of eigenvalue problems or their generalisations;through such methods it is often possible to design algebraic data called “spectral data” whichcharacterize an individual periodic solution of the underlying diﬀerential equation uniquely. In an
infinite-dimensional integrable system, we expect infinitely many spectral quantities, and questionsof convergence and asymptotic behavior plays a major role in their investigation.
It is a remarkable feature in the study of integrable systems that there is very little general theory
that applies to all integrable systems. This is partially due to the vague nature of the concept of an
integrable system. For example, with respect to spectral theory there is no unified definition of theassociated objects that “works” in all integrable systems. Rather, their definition needs to be adapted
to the individual integrable system under investigation. Although the results one obtains in this way
for the various integrable systems are very similar to one another, because of these diﬀerences there
does not exist a unified proof of these results. Rather, they need to be proven for each integrable
system separately.
For this reason we will discuss three individual integrable systems and the associated spectraltheory.
1. The Korteweg-de Vries (KdV) equation is the partial diﬀerential equation for a functionu = u(x, t) in two real variables
ut + uxxx − 6 u ux = 0 .
This equation is named for the two Dutch mathematicians Korteweg and de Vries, who studied
it as a mathematical model of waves on a shallow water surface. The associated integrable
system is the one which sparked the interest in integrable systems, for which most of theaspects of integrable systems theory were first discovered, and which probably remains themost studied infinite-dimensional integrable system today. In particular, in 1968, P. Lax madethe fundamental discovery that the KdV equation can be written in what is today called Laxform, i.e. in the formdL = [B, L] ,(∗)
dt
where L and B are two diﬀerential operators depending on the solution u . For the KdV
equation, L is the 1-dimensional Schrödinger operator
L=−1d2+u,dx2

H. J. Hitchin et al., Integrable Systems, Oxford 1999, p. 1

andd3dd
B = 4 3 − 3u
−3 u.
dx
dx
dx
It is known that the periodic spectrum of the operator L = L(t) is a pure point spectrum,and therefore equal to the set of periodic eigenvalues. We will see that the spectrum of L(t)is conserved in the sense that it does not depend on t . We will use it to construct “spectraldata” for periodic solutions of the KdV equation.

The second order diﬀerential operator L can be rewritten as a 2-dimensional first order diﬀerd
ential operator, and then the eigenvalue equation Lψ = λψ takes the form dx
Ψ = Uλ Ψ witha 2-dimensional periodic operator Uλ depending on the eigenvalue λ . There moreover exists
another 2-dimensional periodic operator Vλ (also depending on λ ) so that the Lax equation(∗) is equivalent to the zero curvature condition∂Uλ ∂Vλ−+ [Uλ , Vλ ] = 0 .
∂t
∂x(†)

2. The 2-dimensional (self-focusing) non-linear Schrödinger equation (sfNLS) is the partial
diﬀerential equationuxx − 2i ut + 2|u|2 u = 0 .
Its original significance comes from the physics of quantum mechanics, but it also plays a rolein the reconstruction of a curve in a 3-dimensional space form from its (Hasimoto) complex
curvature. This diﬀerential equation can also be written in Lax form, which gives rise to aspectral theory in a similar way as with the KdV equation.
3. The 2-dimensional sinh-Gordon equation is the partial diﬀerential equation△u + sinh(u) = 0 .
An immersed surface in a 3-dimensional space form without umbilic points has constant mean
curvature if and only if near every point of the surface, there exists a conformal coordinate z sothat the conformal factor u of the induced metric eu/2 dz dz is a solution of the sinh-Gordon
equation. This partial diﬀerential equation can not be written in Lax form. However, the zerocurvature condition (†) does generalize to the integrable system of the sinh-Gordon equation.
Although it can no longer be interpreted as an eigenvalue equation for some diﬀerential operator(because the dependency of the operators Uλ , Vλ on λ is more complicated), we will see thatit gives rise to a theory that is analogous to the spectral theory of the preceding two cases, andwe will take the liberty of applying the label of “spectral theory” also here.

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