%0 Journal Article
%T An Integrable Equation Governing Short Waves in a Long-wave Model
%+ Laboratoire de Physique Théorique et Astroparticules (LPTA)
%A Faquir, Mohamed
%A Manna, Miguel
%A Neveu, André
%< avec comité de lecture
%Z 07-016; 07-016
%@ 1364-5021
%J Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
%I Royal Society, The
%V 463
%N 2084
%P 1939-1954
%8 2007
%D 2007
%R 10.1098/rspa.2007.1861
%K short-wave dynamics
%K integrable systems
%K Lax pair
%K perturbation theory
%K finite-time singularities
%Z Physics [physics]/Mathematical Physics [math-ph]Journal articles
%X 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.
%G English
%L hal-00266216
%U https://hal.science/hal-00266216
%~ IN2P3
%~ LPTA
%~ CNRS
%~ UNIV-MONTP2
%~ TDS-MACS
%~ UNIV-MONTPELLIER
%~ UM1-UM2