The purpose of this paper is to study the spatially periodic case of the shallow water equation $$\partial v/\partial t+ v\partial t/\partial x+\partial p/\partial x= 0\tag 1$$ in which the “pressure” $p$ is $(1- d^2/dx^2)^{-1}(v^2+{1\over 2} v^{\prime 2})$. The equation has been much studied in recent years, starting with {\it R. Camma} and {\it D. D. Holm} [Phys. Rev. Lett. 81, 1661-1664 (1993)] and {\it R. Camassa}, {\it D. D. Holm} and {\it J. M. Hyman} [Adv. Appl. Math. 31, 1-33 (1994;

Zbl 0808.76011)]. It is notable for a) its complete integrability, b) the existence of peaked solitons and the simplicity of their superposition, c) the presence of breaking waves (blow-up), and d) the equivalence of the flow to the geodesic flow on the group of (compressible) diffeomorphisms of the line with its $H^1$-type geometry.
The equation is similar to KdV but different in many details, as is its mode of solution. There is an associated “acoustic” spectral problem $$y''+\textstyle{{1\over 4}} y=\lambda my\tag 2$$ in which $m= v- v''$, and it is natural to attempt the linearization of (1) on the Jacobi variety of the (hyperelliptic) Riemann surface associated with the Floquet multipliers of (2). This does not work well, but a double cover of that Riemann surface does the trick. That is one novelty compared to the case of KdV. Another is the appearance on the Riemann surface of some number of nodes where KdV would have ordinary ramifications. These correspond to the possibility of “soliton-anti-soliton collisions” and have to be treated in a novel way.

Reviewer: H.P.McKean (New York) and A.Constantin (Zürich)