TY - JOUR

T1 - Transversally periodic solitary gravity-capillary waves

AU - Milewski, Paul A.

AU - Wang, Zhan

PY - 2014/1

Y1 - 2014/1

N2 - When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two- and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity-capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two- and three-dimensional computations of fully localized gravity-capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

AB - When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two- and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity-capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two- and three-dimensional computations of fully localized gravity-capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

UR - http://www.scopus.com/inward/record.url?scp=84890102568&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1098/rspa.2013.0537

U2 - 10.1098/rspa.2013.0537

DO - 10.1098/rspa.2013.0537

M3 - Article

AN - SCOPUS:84890102568

VL - 470

JO - Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences

SN - 1364-503X

IS - 2161

M1 - 20130537

ER -