A new integrable shallow water equation. (English) Zbl 0808.76011

Adv. Appl. Mech. 31, 1-33 (1994).
From the introduction: We discuss a newly discovered completely integrable dispersive shallow water equation \[ u_ t+ 2\kappa u_ x- u_{xxt}+ 3uu_ x= 2u_ x u_{xx}+ uu_{xxx},\tag{1} \] 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.
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.
For the entire collection see [Zbl 0799.00015].


76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology