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Zbl 0808.76011
Camassa, Roberto; Holm, Darryl D.; Hyman, James M.
A new integrable shallow water equation. (English)
[A] Adv. Appl. Mech. 31, 1-33 (1994). ISBN 0-12-002031-9/hbk

From the introduction: We discuss a newly discovered completely integrable dispersive shallow water equation $$u\sb t+ 2\kappa u\sb x- u\sb{xxt}+ 3uu\sb x= 2u\sb x u\sb{xx}+ uu\sb{xxx},\tag 1$$ where $u$ is the fluid velocity in the $x$ direction (or equivalently, the height of the water's free surface above a flat bottom), $\kappa$ is a constant related to the critical shallow- water wave speed.\par After briefly discussing the Boussinesq class of equations for small amplitude dispersive shallow water equations, in Section II we derive the one-dimensional Green-Naghdi equations. In Section III, we use Hamiltonian methods to obtain equation (1) for unidirectional waves. In Section IV, we analyze the behavior of the solutions of (1) and show that certain initial conditions develop a vertical slope in finite time. We also show that there exist stable multisoliton solutions and derive the phase shift that occurs when two of these solitons collide. Section V demonstrates the existence of an infinite number of conservation laws for equation (1) that follow from its bi-Hamiltonian property. Section VI uses this property to derive the isospectral problem for this equation and others in its hierarchy.

MSC 2000:: *76B15 Wave motions (fluid mechanics)
76B25 Solitary waves, etc. (inviscid fluids)
35Q51 Solitons
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology

Keywords: Boussinesq class of equations; one-dimensional Green-Naghdi equations; Hamiltonian methods; vertical slope; stable multisoliton solutions; phase shift; conservation laws; bi-Hamiltonian property; isospectral problem

Cited in: Zbl 0940.35177 Zbl 0952.35114

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