zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A shallow water equation on the circle. (English) Zbl 0940.35177

The purpose of this paper is to study the spatially periodic case of the shallow water equation

v/t+vt/x+p/x=0(1)

in which the “pressure” p is (1-d 2 /dx 2 ) -1 (v 2 +1 2v '2 ). The equation has been much studied in recent years, starting with R. Camma and D. D. Holm [Phys. Rev. Lett. 81, 1661-1664 (1993)] and R. Camassa, D. D. Holm and 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 '' +1 4y=λmy(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.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
37K20Relations of infinite-dimensional systems with algebraic geometry, etc.
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction