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Mathematics of dispersive water waves. (English) Zbl 1093.37511
Summary: A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.
MSC:
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q30Stokes and Navier-Stokes equations
37L30Attractors and their dimensions, Lyapunov exponents
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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