American Institute of Mathematical Sciences

Advanced
October  2009, 12(3): 607-621. doi: 10.3934/dcdsb.2009.12.607

Nonlinear resonances of water waves

Elena Kartashova 1,

1. 

RISC, J. Kepler University, Altenbergerstr. 69, Linz, A-4040, Austria

Received  April 2009 Revised  June 2009 Published  July 2009

In the last fifteen years great progress has been made in the understanding of nonlinear resonance dynamics of water waves. Notions of scale- and angle-resonances have been introduced, new type of energy cascade due to nonlinear resonances in the gravity water waves has been discovered, conception of a resonance cluster has been much and successfully employed, a novel model of laminated wave turbulence has been developed, etc. etc. Two milestones in this area of research have to be mentioned: a) development of the $q$-class method which is effective for computing integer points on resonance manifolds, and b) construction of marked planar graphs, instead of classical resonance curves, representing simultaneously all resonance clusters in a finite spectral domain, together with their dynamical systems. Among them, new integrable dynamical systems have been found that can be used for explaining numerical and laboratory results. The aim of this paper is to give a brief overview of our current knowledge about nonlinear resonances among water waves, and finally to formulate the three most important open problems.
Keywords: Nonlinear resonances, dynamical systems, fluid mechanics..
Mathematics Subject Classification: Primary: 74J30, 37N10; Secondary: 37-0.
Citation: Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607
[1]

François Gay-Balmaz, Darryl D. Holm. Predicting uncertainty in geometric fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-14. doi: 10.3934/dcdss.2020071
[2]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233
[3]

Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
[4]

Alain Miranville, Mazen Saad, Raafat Talhouk. Preface: Workshop in fluid mechanics and population dynamics. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : i-i. doi: 10.3934/dcdss.2014.7.2i
[5]

Eduard Feireisl, Šárka Nečasová, Reimund Rautmann, Werner Varnhorn. New developments in mathematical theory of fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : i-ii. doi: 10.3934/dcdss.2014.7.5i
[6]

Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165
[7]

Mahdi Khajeh Salehani. Identification of generic stable dynamical systems taking a nonlinear differential approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4541-4555. doi: 10.3934/dcdsb.2018175
[8]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469
[9]

Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091
[10]

Angelo Morro. Nonlinear diffusion equations in fluid mixtures. Evolution Equations & Control Theory, 2016, 5 (3) : 431-448. doi: 10.3934/eect.2016012
[11]

Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829
[12]

Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893
[13]

Caixia Gao, Enmin Feng, Zongtao Wang, Zhilong Xiu. Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls. Journal of Industrial & Management Optimization, 2005, 1 (3) : 377-388. doi: 10.3934/jimo.2005.1.377
[14]

Masatoshi Shiino, Keiji Okumura. Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena. Conference Publications, 2013, 2013 (special) : 685-694. doi: 10.3934/proc.2013.2013.685
[15]

Brian Straughan. Shocks and acceleration waves in modern continuum mechanics and in social systems. Evolution Equations & Control Theory, 2014, 3 (3) : 541-555. doi: 10.3934/eect.2014.3.541
[16]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[17]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[18]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[19]

John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361
[20]

Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences