zbMATH — the first resource for mathematics

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 \[ \partial v/\partial t+ v\partial t/\partial x+\partial p/\partial x= 0\tag{1} \] 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 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 \[ y''+\textstyle{{1\over 4}} y=\lambda my\tag{2} \] 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.

35Q53 KdV equations (Korteweg-de Vries equations)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Albers, Lett Math Phys 32 pp 137– (1994) · Zbl 0808.35124
[2] Albers, Proc Roy Soc London A 450 pp 677– (1995) · Zbl 0835.35125
[3] Arnold, Ann de l’Inst Fourier 16 pp 319– (1966) · Zbl 0148.45301
[4] Mathematical Methods of Classical Mechanics. Springer-Verlag, New York/Heidelberg/Berlin 1978.
[5] Atkinson, J Reine Angew Math 375/6 pp 380– (1987)
[6] Camassa, Phys Rev Letters 81 pp 1661– (1993) · Zbl 0972.35521
[7] Camassa, Adv Appl Mech 31 pp 1– (1994)
[8] Constantin, Comm Pure Appl Math 51 pp 475– (1998) · Zbl 0934.35153
[9] Ebin, Ann Math 92 pp 102– (1970) · Zbl 0211.57401
[10] Fokas, Physical 4 D pp 47– (1981)
[11] Hochstadt, Math Zeit 82 pp 237– (1963) · Zbl 0127.04203
[12] McKean, Comm Pure Appl Math 30 pp 347– (1977) · Zbl 0335.58013
[13] McKean, Comm Pure Appl Math 29 pp 143– (1976) · Zbl 0339.34024
[14] McKean, Invent Math 30 pp 217– (1975) · Zbl 0319.34024
[15] Misiolek, J Geom Phys 24 pp 203– (1998) · Zbl 0901.58022
[16] Novikov, Func Anal Appl 8 pp 54– (1974) · Zbl 0301.54027
[17] Trubowitz, Comm Pure Appl Math 30 pp 321– (1977) · Zbl 0403.34022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.