scholarly journals A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

2017 ◽  
Vol 228 (3) ◽  
pp. 773-820 ◽  
Author(s):  
Boris Buffoni ◽  
Mark D. Groves ◽  
Erik Wahlén
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Water Wave ◽  
Three Dimensional ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Weak Surface

scholarly journals A remark on the two dimensional water wave problem with surface tension

2019 ◽  
Vol 266 (9) ◽  
pp. 5748-5771
Author(s):  
Shuanglin Shao ◽  
Hsi-Wei Shih
Keyword(s):  
Surface Tension ◽  
Water Wave ◽  
Two Dimensional ◽  
Wave Problem ◽  
Water Wave Problem

The NLS and FWI approximation make wrong predictions for the water wave problem in case of small surface tension

PAMM ◽  
2014 ◽  
Vol 14 (1) ◽  
pp. 747-748
Author(s):  
Danish Ali Sunny ◽  
Guido Schneider ◽  
Dominik Zimmermann
Keyword(s):  
Surface Tension ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Small Surface

scholarly journals Local smoothing effects for the water-wave problem with surface tension

2009 ◽  
Vol 347 (3-4) ◽  
pp. 159-162 ◽  
Author(s):  
Hans Christianson ◽  
Vera Mikyoung Hur ◽  
Gigliola Staffilani
Keyword(s):  
Surface Tension ◽  
Water Wave ◽  
Local Smoothing ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Smoothing Effects

scholarly journals A global investigation of solitary-wave solutions to a two-parameter model for water waves

1997 ◽  
Vol 342 ◽  
pp. 199-229 ◽  
Author(s):  
ALAN R. CHAMPNEYS ◽  
MARK D. GROVES
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Model Equation ◽  
Global Bifurcation ◽  
Solitary Wave Solutions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Wave Solutions

The model equationformula herearises as the equation for solitary-wave solutions to a fifth-order long-wave equation for gravity–capillary water waves. Being Hamiltonian, reversible and depending upon two parameters, it shares the structure of the full steady water-wave problem. Moreover, all known analytical results for local bifurcations of solitary-wave solutions to the full water-wave problem have precise counterparts for the model equation.At the time of writing two major open problems for steady water waves are attracting particular attention. The first concerns the possible existence of solitary waves of elevation as local bifurcation phenomena in a particular parameter regime; the second, larger, issue is the determination of the global bifurcation picture for solitary waves. Given that the above equation is a good model for solitary waves of depression, it seems natural to study the above issues for this equation; they are comprehensively treated in this article.The equation is found to have branches of solitary waves of elevation bifurcating from the trivial solution in the appropriate parameter regime, one of which is described by an explicit solution. Numerical and analytical investigations reveal a rich global bifurcation picture including multi-modal solitary waves of elevation and depression together with interactions between the two types of wave. There are also new orbit-flip bifurcations and associated multi-crested solitary waves with non-oscillatory tails.

scholarly journals Transverse instability of solitary-wave states of the water-wave problem

2001 ◽  
Vol 439 ◽  
pp. 255-278 ◽  
Author(s):  
T. J. BRIDGES
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Explicit Calculation ◽  
Geometric Condition ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Transverse Instability ◽  
Stability And Instability

Transverse stability and instability of solitary waves correspond to a class of perturbations that are travelling in a direction transverse to the direction of the basic solitary wave. In this paper we consider the problem of transverse instability of solitary waves for the water-wave problem, from both the model equation point of view and the full water-wave equations. A new universal geometric condition for transverse instability forms the backbone of the analysis. The theory is first illustrated by application to model PDEs for water waves such as the KP equation, and then it is applied to the full water-wave problem. This is the first theory proposed for transverse instability of solitary waves of the full water-wave problem. The theory suggests the introduction of a new functional for water waves, whose importance is suggested by the mathematical structure. Without explicit calculation, the theory is used to argue that the basic class of solitary waves of the water-wave problem, which bifurcate at Froude number unity, are likely to be stable to transverse perturbations, even at large amplitude.

scholarly journals Some Explicit Solutions to the Three-Dimensional Nonlinear Water Wave Problem

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Calin I. Martin
Keyword(s):  
New York ◽  
Surface Pressure ◽  
Water Wave ◽  
Three Dimensional ◽  
Wave Climate ◽  
Explicit Solutions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Constant Vorticity ◽  
Radial Structure

AbstractWe present some explicit solutions (given in Eulerian coordinates) to the three-dimensional nonlinear water wave problem. The velocity field of some of the solutions exhibits a non-constant vorticity vector. An added bonus of the solutions we find is the possibility of incorporating a variable (in time and space) surface pressure which has a radial structure. A special type of radial structure of the surface pressure (of exponential type) is one of the features displayed by hurricanes, cf. Overland (Earle, Malahoff (eds) Overland in ocean wave climate, Plenum Pub. Corp., New York, 1979).

Sign in / Sign up

Export Citation Format

Share Document