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Lax pairs and higher order models for water waves. (English) Zbl 0796.35148

It is shown a local well-posedness result (in \(H^ s(\mathbb{R})\), \(s \geq 4)\) for a class of higher order nonlinear dispersive equations. The analysis is restricted to the model \[ u_ t + c_ 1 u_ xu_{xx} + c_ 2uu_{xxx} + c_ 3u_{xxxxx} = 0 \] which resembles the essential features of the next equation in the KdV hierarchy (beyond the KdV itself) and some models appearing in water wave problems.
Reviewer: L.Vazquez (Madrid)

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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