scholarly journals Weakly Nonlinear Theory of Pattern-Forming Systems with Spontaneously Broken Isotropy

1996 ◽  
Vol 76 (25) ◽  
pp. 4729-4732 ◽  
Author(s):  
A. G. Rossberg ◽  
A. Hertrich ◽  
L. Kramer ◽  
W. Pesch
Keyword(s):  
Nonlinear Theory ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Pattern Forming

scholarly journals Can weakly nonlinear theory explain Faraday wave patterns near onset?

2015 ◽  
Vol 777 ◽  
pp. 604-632 ◽  
Author(s):  
A. C. Skeldon ◽  
A. M. Rucklidge
Keyword(s):  
Nonlinear Theory ◽  
Stokes Equations ◽  
Single Mode ◽  
Theoretical Work ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Temporal Modes ◽  
Pattern Forming

The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode, and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free-boundary Navier–Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier–Stokes equations and (previously published) experimental results for the Faraday problem with multiple-frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stabilised, but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure.

scholarly journals Weakly nonlinear theory and sea state bias estimations

1999 ◽  
Vol 104 (C4) ◽  
pp. 7641-7647 ◽  
Author(s):  
Tanos Elfouhaily ◽  
Donald Thompson ◽  
Douglas Vandemark ◽  
Bertrand Chapron
Keyword(s):  
Nonlinear Theory ◽  
Sea State ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Sea State Bias

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.

Extended weakly nonlinear theory of planar nematic convection

1999 ◽  
Vol 59 (2) ◽  
pp. 1747-1769 ◽  
Author(s):  
Emmanuel Plaut ◽  
Werner Pesch
Keyword(s):  
Nonlinear Theory ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory
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