scholarly journals Nonlinear water waves: introduction and overview

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170310 ◽  
Author(s):  
A. Constantin
Keyword(s):  
Water Waves ◽  
Essential Feature ◽  
Theme Issue ◽  
Research Activity ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Isaac Newton ◽  
Weakly Nonlinear Theory ◽  
Mathematical Sciences ◽  
To Come

For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals On three-dimensional Gerstner-like equatorial water waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170088 ◽  
Author(s):  
D. Henry
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Equatorial Region ◽  
Mathematical Research ◽  
Theme Issue ◽  
Research Activity ◽  
Nonlinear Water Waves ◽  
Lagrangian Variables

This paper reviews some recent mathematical research activity in the field of nonlinear geophysical water waves. In particular, we survey a number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave–current interactions, in the equatorial region. These solutions are nonlinear, three-dimensional and explicit in terms of Lagrangian variables. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Nonlinear water waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1501-1504 ◽  
Author(s):  
Adrian Constantin
Keyword(s):  
Water Waves ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Recent Developments

This introduction to the issue provides a review of some recent developments in the study of water waves. The content and contributions of the papers that make up this Theme Issue are also discussed.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals On the short-wavelength stabilities of some geophysical flows

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170090 ◽  
Author(s):  
Delia Ionescu-Kruse
Keyword(s):  
Water Waves ◽  
Short Wavelength ◽  
Travelling Wave ◽  
Geophysical Flows ◽  
Rotating Flows ◽  
Steady Flows ◽  
Azimuthal Direction ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Stability Method

This paper is a survey of the short-wavelength stability method for rotating flows. Additional complications such as stratification in the flow or the presence of non-conservative body forces are considered too. This method is applied to the specific study of some exact geophysical flows. For Gerstner-like geophysical flows one can identify perturbations in certain directions as a source of instabilities with an exponentially growing amplitude, the growth rate of the instabilities depending on the steepness of the travelling wave profile. On the other hand, for certain physically realistic velocity profiles, steady flows moving only in the azimuthal direction, with no variation in this direction, are locally stable to the short-wavelength perturbations. This article is part of the theme issue ‘Nonlinear water waves’.

Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant

1991 ◽  
Vol 229 (-1) ◽  
pp. 135 ◽  
Author(s):  
S. W. Joo ◽  
A. F. Messiter ◽  
W. W. Schultz
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear

scholarly journals Water waves, nonlinear Schrödinger equations and their solutions

1983 ◽  
Vol 25 (1) ◽  
pp. 16-43 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Water Waves ◽  
Rational Solution ◽  
Three Dimensional ◽  
Length Scale ◽  
Nls Equation ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Schrödinger ◽  
Wave Induced ◽  
Modulation Length

AbstractEquations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

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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