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An integrable equation governing short waves in a long-wave model

2007; Royal Society; Volume: 463; Issue: 2084 Linguagem: Inglês

10.1098/rspa.2007.1861

1471-2946

Mohamed Faquir, M. A. Manna, A. Neveu,

Ocean Waves and Remote Sensing

The dynamics of a nonlinear and dispersive long surface capillary-gravity wave model equation is studied analytically in its short-wave limit. We exhibit its Lax pair and some non-trivial conserved quantities. Through a change of functions, an unexpected connection between this classical surface water-wave model and the sine-Gordon (or sinh-Gordon) equation is established. Numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time. These features can be traced to the fact that the surface tension term in the energy involves second-order derivatives. It would be interesting to see in an experiment whether such singularities actually appear, for which surface tension would be specifically responsible.