Global bifurcation for water waves with capillary effects and constant vorticity

2013 ◽  
Vol 174 (3) ◽  
pp. 459-475 ◽  
Author(s):  
Bogdan-Vasile Matioc
Keyword(s):  
Water Waves ◽  
Global Bifurcation ◽  
Capillary Effects ◽  
Constant Vorticity

Existence of Wilton Ripples for Water Waves with Constant Vorticity and Capillary Effects

2013 ◽  
Vol 73 (4) ◽  
pp. 1582-1595 ◽  
Author(s):  
Calin Iulian Martin ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Water Waves ◽  
Capillary Effects ◽  
Constant Vorticity

scholarly journals A non-local formulation of rotational water waves

2011 ◽  
Vol 689 ◽  
pp. 129-148 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Velocity Potential ◽  
Vries Equation ◽  
Travelling Waves ◽  
Explicit Dependence ◽  
Local Equation ◽  
Constant Vorticity ◽  
Small Parameters ◽  
Non Local

AbstractThe classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.

scholarly journals Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

2015 ◽  
Vol 145 (4) ◽  
pp. 791-883 ◽  
Author(s):  
M. D. Groves ◽  
E. Wahlén
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Finite Depth ◽  
Constant Vorticity ◽  
Existence And Stability ◽  
Strong Surface ◽  
Weak Surface ◽  
The Stability ◽  
The Waves

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy𝓗subject to the constraint𝓘= 2µ, where𝓘is the wave momentum and 0 <µ≪ 1. Since𝓗and𝓘are both conserved quantities, a standard argument asserts the stability of the setDµof minimizers: solutions starting nearDµremain close toDµin a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation asµ↓ 0.

scholarly journals Existence of capillary-gravity water waves with piecewise constant vorticity

2014 ◽  
Vol 256 (8) ◽  
pp. 3086-3114 ◽  
Author(s):  
Calin Iulian Martin ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Water Waves ◽  
Piecewise Constant ◽  
Constant Vorticity

scholarly journals Analytical approximation and numerical simulations for periodic travelling water waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170093 ◽  
Author(s):  
Konstantinos Kalimeris
Keyword(s):  
Water Waves ◽  
Optimization Problem ◽  
Analytical Approximation ◽  
Numerical Results ◽  
Numerical Continuation ◽  
Constrained Optimization Problem ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Constant Vorticity ◽  
Continuation Algorithm

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity. This article is part of the theme issue ‘Nonlinear water waves’.

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