On the stability of lumps and wave collapse in water waves

Cited 22 time in webofscience Cited 0 time in scopus
  • Hit : 296
  • Download : 0
In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg de Vries solitary waves, the present study is concerned with a distinct class of gravity - capillary lumps recently found on water of finite or in finite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution ( ground state) of an elliptic equation system of the Benney - Roskes - Davey - Stewartson ( BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev Petviashvili equation, a model for weakly nonlinear gravity - capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.
Publisher
ROYAL SOC
Issue Date
2008-08
Language
English
Article Type
Article; Proceedings Paper
Keywords

CAPILLARY SOLITARY WAVES; GRAVITY-WAVES; SURFACE-WAVES; DEEP-WATER; PACKETS; EQUATIONS; FLOWS

Citation

PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, v.366, no.1876, pp.2761 - 2774

ISSN
1364-503X
DOI
10.1098/rsta.2008.0047
URI
http://hdl.handle.net/10203/93261
Appears in Collection
ME-Journal Papers(저널논문)
Files in This Item
There are no files associated with this item.
This item is cited by other documents in WoS
⊙ Detail Information in WoSⓡ Click to see webofscience_button
⊙ Cited 22 items in WoS Click to see citing articles in records_button

qr_code

  • mendeley

    citeulike


rss_1.0 rss_2.0 atom_1.0