TY - JOUR

T1 - On weakly nonlinear gravity-capillary solitary waves

AU - Kim, Boguk

AU - Dias, Frédéric

AU - Milewski, Paul A.

N1 - Funding Information:
The authors wish to thank Prof. T.R. Akylas for his helpful comments in clarifying a particular meaning of this model equations system in the quantitative perspective. This paper was partially supported by the French Agence Nationale de la Recherche project MANUREVA ANR-08-SYSC-019 and by Priority Research Centers Program through the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science and Technology of the Korean government (2010-0029638) . PAM wishes to thank the CNRS for sponsoring his visit to Ecole Normale Supérieure de Cachan in 2008 2009, the NSF for support under Grant DMS-0908077 , and the EPSRC for support under Grant Number GR/S47786/01 .

PY - 2012/3

Y1 - 2012/3

N2 - As a weakly nonlinear model equations system for gravity-capillary waves on the surface of a potential fluid flow, a cubic-order truncation model is presented, which is derived from the ordinary Taylor series expansion for the free boundary conditions of the Euler equations with respect to the velocity potential and the surface elevation. We assert that this model is the optimal reduced simplified model for weakly nonlinear gravity-capillary solitary waves mainly because the generation mechanism of weakly nonlinear gravity-capillary solitary waves from this model is consistent with that of the full Euler equations, both quantitatively and qualitatively, up to the third order in amplitude. In order to justify our assertion, we show that this weakly nonlinear model in deep water allows gravity-capillary solitary wavepackets in the weakly nonlinear and narrow bandwidth regime where the classical nonlinear Schrödinger (NLS) equation governs; this NLS equation derived from the model is identical to the one directly derived from the Euler equations. We verify that both quantitative and qualitative properties of the gravity-capillary solitary waves of the model precisely agree with the counterparts of the Euler equations near the bifurcation point by performing a numerical continuation to find the steady profiles of weakly nonlinear gravity-capillary solitary waves of the primary stable bifurcation branch. In addition, unsteady numerical simulations, in which those solitary waves are used as initial conditions, are provided as supporting evidences.

AB - As a weakly nonlinear model equations system for gravity-capillary waves on the surface of a potential fluid flow, a cubic-order truncation model is presented, which is derived from the ordinary Taylor series expansion for the free boundary conditions of the Euler equations with respect to the velocity potential and the surface elevation. We assert that this model is the optimal reduced simplified model for weakly nonlinear gravity-capillary solitary waves mainly because the generation mechanism of weakly nonlinear gravity-capillary solitary waves from this model is consistent with that of the full Euler equations, both quantitatively and qualitatively, up to the third order in amplitude. In order to justify our assertion, we show that this weakly nonlinear model in deep water allows gravity-capillary solitary wavepackets in the weakly nonlinear and narrow bandwidth regime where the classical nonlinear Schrödinger (NLS) equation governs; this NLS equation derived from the model is identical to the one directly derived from the Euler equations. We verify that both quantitative and qualitative properties of the gravity-capillary solitary waves of the model precisely agree with the counterparts of the Euler equations near the bifurcation point by performing a numerical continuation to find the steady profiles of weakly nonlinear gravity-capillary solitary waves of the primary stable bifurcation branch. In addition, unsteady numerical simulations, in which those solitary waves are used as initial conditions, are provided as supporting evidences.

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U2 - 10.1016/j.wavemoti.2011.10.002

DO - 10.1016/j.wavemoti.2011.10.002

M3 - Article

AN - SCOPUS:84856562975

SN - 0165-2125

VL - 49

SP - 221

EP - 237

JO - Wave Motion

JF - Wave Motion

IS - 2

ER -