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Two-component integrable systems modelling shallow water waves: the constant vorticity case. (English) Zbl 1231.76040
Summary: We describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of mass conservation, the simplest bottom and surface conditions and the constant vorticity condition. The approximate model equations are generated by introduction of suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order. The so obtained equations can be related to three different integrable systems: a two component generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system.
MSC:
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35PDEs in connection with fluid mechanics
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37N10Dynamical systems in fluid mechanics, oceanography and meteorology
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