# Discrete resonances in nonlinear wavesystems [DIRNOW]

### Project Lead

### Project Duration

01/10/2007 - 30/10/2010### Project URL

Go to Website## Publications

### 2011

[Schreiner]

### Software

#### Wolfgang Schreiner

In: Nonlinear Resonance Analysis - Theory, Computation, Applications, Elena Kartashova (ed.), pp. 185-208. 2011. Cambridge University Press, Cambridge, UK, ISBN 978-0-521-76360-8. Appendix.@

author = {Wolfgang Schreiner},

title = {{Software}},

booktitle = {{Nonlinear Resonance Analysis - Theory, Computation, Applications}},

language = {english},

pages = {185--208},

publisher = {Cambridge University Press},

address = {Cambridge, UK},

isbn_issn = {ISBN 978-0-521-76360-8},

year = {2011},

note = {Appendix},

editor = {Elena Kartashova},

refereed = {yes},

length = {24}

}

**incollection**{RISC4164,author = {Wolfgang Schreiner},

title = {{Software}},

booktitle = {{Nonlinear Resonance Analysis - Theory, Computation, Applications}},

language = {english},

pages = {185--208},

publisher = {Cambridge University Press},

address = {Cambridge, UK},

isbn_issn = {ISBN 978-0-521-76360-8},

year = {2011},

note = {Appendix},

editor = {Elena Kartashova},

refereed = {yes},

length = {24}

}

### 2010

[Kartaschova]

### Discrete wave turbulence of rotational capillary water waves

#### A. Constantin, E. Kartashova, E. Wahlen

Phys. Fluids submitted, pp. 1-13. 2010. AIP, isbn. [pdf]@

author = {A. Constantin and E. Kartashova and E. Wahlen},

title = {{Discrete wave turbulence of rotational capillary water waves}},

language = {english},

abstract = {We study the discrete wave turbulent regime of capillarywater waves with constant non-zero vorticity. The explicitHamiltonian formulation and the corresponding couplingcoefficient are obtained. We also present the constructionand investigation of resonance clustering. Some physicalimplications of the obtained results are discussed.},

journal = {Phys. Fluids},

volume = {submitted},

pages = {1--13},

publisher = {AIP},

isbn_issn = {isbn},

year = {2010},

refereed = {yes},

length = {13}

}

**article**{RISC3932,author = {A. Constantin and E. Kartashova and E. Wahlen},

title = {{Discrete wave turbulence of rotational capillary water waves}},

language = {english},

abstract = {We study the discrete wave turbulent regime of capillarywater waves with constant non-zero vorticity. The explicitHamiltonian formulation and the corresponding couplingcoefficient are obtained. We also present the constructionand investigation of resonance clustering. Some physicalimplications of the obtained results are discussed.},

journal = {Phys. Fluids},

volume = {submitted},

pages = {1--13},

publisher = {AIP},

isbn_issn = {isbn},

year = {2010},

refereed = {yes},

length = {13}

}

[Kartaschova]

### Capillary freak waves in He-II as a manifestation of discrete wave turbulent regime

#### E. Kartashova

In: Geophysical Research Abstracts, E. Pelinovsky, C. Kharif (ed.), Proceedings of EGU 2010 (European Geosciences Union, General Assembly 2010),12, pp. 1889-1889. 2010. issn. [pdf]@

author = {E. Kartashova},

title = {{Capillary freak waves in He-II as a manifestation of discrete wave turbulent regime}},

booktitle = {{Geophysical Research Abstracts}},

language = {english},

abstract = {Two fundamental findings of the modern theory of wave turbulence are 1) existence of Kolmogorov-Zakharov power energy spectra (KZ-spectra) in $k$-space, \cite{zak2}, and 2) existence of ``gaps" in KZ-spectra corresponding to the resonance clustering, \cite{K06-1}. Accordingly, three wave turbulent regimes can be singled out - \emph{kinetic} (described by wave kinetic equations and KZ-spectra, in random phase approximation, \cite{ZLF92}); \emph{discrete} (described by a few dynamical systems, with coherent phases corresponding to resonance conditions, \cite{K09b}) and \emph{mesoscopic} (where kinetic and discrete evolution of the wave field coexist, \cite{zak4}).We present an explanation of freak waves appearance in capillary waves in He-II, \cite{ABKL09}, as a manifestation of discrete wave turbulent regime. Implications of these results for other wave systems are briefly discussed.},

volume = {12},

pages = {1889--1889},

isbn_issn = {issn},

year = {2010},

editor = { E. Pelinovsky and C. Kharif },

refereed = {yes},

length = {1},

conferencename = {EGU 2010 (European Geosciences Union, General Assembly 2010),}

}

**inproceedings**{RISC3933,author = {E. Kartashova},

title = {{Capillary freak waves in He-II as a manifestation of discrete wave turbulent regime}},

booktitle = {{Geophysical Research Abstracts}},

language = {english},

abstract = {Two fundamental findings of the modern theory of wave turbulence are 1) existence of Kolmogorov-Zakharov power energy spectra (KZ-spectra) in $k$-space, \cite{zak2}, and 2) existence of ``gaps" in KZ-spectra corresponding to the resonance clustering, \cite{K06-1}. Accordingly, three wave turbulent regimes can be singled out - \emph{kinetic} (described by wave kinetic equations and KZ-spectra, in random phase approximation, \cite{ZLF92}); \emph{discrete} (described by a few dynamical systems, with coherent phases corresponding to resonance conditions, \cite{K09b}) and \emph{mesoscopic} (where kinetic and discrete evolution of the wave field coexist, \cite{zak4}).We present an explanation of freak waves appearance in capillary waves in He-II, \cite{ABKL09}, as a manifestation of discrete wave turbulent regime. Implications of these results for other wave systems are briefly discussed.},

volume = {12},

pages = {1889--1889},

isbn_issn = {issn},

year = {2010},

editor = { E. Pelinovsky and C. Kharif },

refereed = {yes},

length = {1},

conferencename = {EGU 2010 (European Geosciences Union, General Assembly 2010),}

}

[Kartaschova]

### Towards a Theory of Discrete and Mesoscopic Wave Turbulence

#### E.Kartashova, V. Lvov, S. Nazarenko, I. Procaccia

Submitted to the RISC Report Series. Technical report no. 10-04 in RISC Report Series, February 2010. [pdf]@

author = {E.Kartashova and V. Lvov and S. Nazarenko and I. Procaccia},

title = {{Towards a Theory of Discrete and Mesoscopic Wave Turbulence}},

language = {english},

abstract = {This is WORK IN PROGRESS carried out in years 2008-2009 and partly supported by Austrian FWF-project P20164-N18 and 6 EU Programme under the project SCIEnce, Contract No. 026133).Abstract:\emph{Discrete wave turbulence} in bounded media refers to the regular and chaotic dynamics of independent (that is, discrete in $k$-space) resonance clusters consisting of finite (often fairly big) number of connected wave triads or quarters, with exact three- or four-wave resonances correspondingly. "Discreteness" means that for small enough amplitudes there is no energy flow among the clusters. Increasing of wave amplitudes and/or of system size opens new channels of wave interactions via quasi-resonant clusters. This changes statistics of energy exchange between waves and results in new, \emph{mesoscopic} regime of \emph{wave turbulence}, where \emph{discrete wave turbulence} and \emph{kinetic wave turbulence} in unbounded media co-exist, the latter well studied in the framework of wave kinetic equations. We overview in systematic manner and from unified viewpoint some preliminary results of studies of regular and stochastic wave behavior in bounded media, aiming to shed light on their relationships and to clarify their role and place in the structure of a future theory of discrete and mesoscopic wave turbulence, elucidated in this paper. We also formulate a set of yet open questions and problems in this new field of nonlinear wave physics, that awaits for comprehensive studies in the framework of the theory. We hope that the resulting theory will offer very interesting issues both from the physical and the methodological viewpoints, with possible important applications in environmental sciences, fluid dynamics, astronomy and plasma physics.},

year = {2010},

month = {February},

howpublished = {Technical report no. 10-04 in RISC Report Series},

length = {42},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

**techreport**{RISC3967,author = {E.Kartashova and V. Lvov and S. Nazarenko and I. Procaccia},

title = {{Towards a Theory of Discrete and Mesoscopic Wave Turbulence}},

language = {english},

abstract = {This is WORK IN PROGRESS carried out in years 2008-2009 and partly supported by Austrian FWF-project P20164-N18 and 6 EU Programme under the project SCIEnce, Contract No. 026133).Abstract:\emph{Discrete wave turbulence} in bounded media refers to the regular and chaotic dynamics of independent (that is, discrete in $k$-space) resonance clusters consisting of finite (often fairly big) number of connected wave triads or quarters, with exact three- or four-wave resonances correspondingly. "Discreteness" means that for small enough amplitudes there is no energy flow among the clusters. Increasing of wave amplitudes and/or of system size opens new channels of wave interactions via quasi-resonant clusters. This changes statistics of energy exchange between waves and results in new, \emph{mesoscopic} regime of \emph{wave turbulence}, where \emph{discrete wave turbulence} and \emph{kinetic wave turbulence} in unbounded media co-exist, the latter well studied in the framework of wave kinetic equations. We overview in systematic manner and from unified viewpoint some preliminary results of studies of regular and stochastic wave behavior in bounded media, aiming to shed light on their relationships and to clarify their role and place in the structure of a future theory of discrete and mesoscopic wave turbulence, elucidated in this paper. We also formulate a set of yet open questions and problems in this new field of nonlinear wave physics, that awaits for comprehensive studies in the framework of the theory. We hope that the resulting theory will offer very interesting issues both from the physical and the methodological viewpoints, with possible important applications in environmental sciences, fluid dynamics, astronomy and plasma physics.},

year = {2010},

month = {February},

howpublished = {Technical report no. 10-04 in RISC Report Series},

length = {42},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

[Kartaschova]

### Resonance clustering in wave turbulent regimes: Integrable dynamics

#### E. Kartashova, M. Bustamante

Physica A: Stat. Mech. Appl. submitted, pp. 1-31. 2010. Elsevier, ISSN: 0378-4371. [pdf]@

author = {E. Kartashova and M. Bustamante},

title = {{Resonance clustering in wave turbulent regimes: Integrable dynamics}},

language = {english},

abstract = {Two fundamental facts of the modern wave turbulence theory are1) existence of power energy spectra in $k$-space, and 2) existence of ``gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic equations and power energy spectra; \emph{discrete}, characterized by resonance clustering; and \emph{mesoscopic},where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Usinga novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters areintegrable in quadratures for arbitrary initial conditions and some others -- only for particular initial conditions. Wealso identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.},

journal = {Physica A: Stat. Mech. Appl.},

volume = {submitted},

pages = {1--31},

publisher = {Elsevier},

isbn_issn = {ISSN: 0378-4371},

year = {2010},

refereed = {yes},

length = {31}

}

**article**{RISC3966,author = {E. Kartashova and M. Bustamante},

title = {{Resonance clustering in wave turbulent regimes: Integrable dynamics}},

language = {english},

abstract = {Two fundamental facts of the modern wave turbulence theory are1) existence of power energy spectra in $k$-space, and 2) existence of ``gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic equations and power energy spectra; \emph{discrete}, characterized by resonance clustering; and \emph{mesoscopic},where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Usinga novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters areintegrable in quadratures for arbitrary initial conditions and some others -- only for particular initial conditions. Wealso identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.},

journal = {Physica A: Stat. Mech. Appl.},

volume = {submitted},

pages = {1--31},

publisher = {Elsevier},

isbn_issn = {ISSN: 0378-4371},

year = {2010},

refereed = {yes},

length = {31}

}

[Kartaschova]

### Turbulence of capillary waves revisited

#### E. Kartashova, A. Kartashov

EPL submitted, pp. 1-6. 2010. isbn. [url]@

author = {E. Kartashova and A. Kartashov},

title = {{Turbulence of capillary waves revisited}},

language = {english},

abstract = {Kinetic regime of capillary wave turbulence is classically regarded in terms of three-wave interactions with the exponent of power energy spectrum being $\nu=-7/4$ (two-dimensional case). We show that a number of assumptions necessary for this regime to occur can not be fulfilled. Four-wave interactions of capillary waves should be taken into account instead, which leads to exponents $\nu=-13/6$ and $\nu=-3/2$ for one- and two-dimensional wavevectors correspondingly. It follows that for general dispersion functions of decay type, three-wave kinetic regime need not prevail and higher order resonances may play a major role.},

journal = {EPL},

volume = {submitted},

pages = {1--6},

isbn_issn = {isbn},

year = {2010},

refereed = {yes},

length = {6},

url = {http://arxiv.org/abs/1005.2067}

}

**article**{RISC4026,author = {E. Kartashova and A. Kartashov},

title = {{Turbulence of capillary waves revisited}},

language = {english},

abstract = {Kinetic regime of capillary wave turbulence is classically regarded in terms of three-wave interactions with the exponent of power energy spectrum being $\nu=-7/4$ (two-dimensional case). We show that a number of assumptions necessary for this regime to occur can not be fulfilled. Four-wave interactions of capillary waves should be taken into account instead, which leads to exponents $\nu=-13/6$ and $\nu=-3/2$ for one- and two-dimensional wavevectors correspondingly. It follows that for general dispersion functions of decay type, three-wave kinetic regime need not prevail and higher order resonances may play a major role.},

journal = {EPL},

volume = {submitted},

pages = {1--6},

isbn_issn = {isbn},

year = {2010},

refereed = {yes},

length = {6},

url = {http://arxiv.org/abs/1005.2067}

}

[Kartaschova]

### Nonlinear Resonance Analysis

#### E. Kartashova

Cambridge edition, 2010. Cambridge University Press, ISBN-13: 9780521763608. [url]@

author = {E. Kartashova},

title = {{Nonlinear Resonance Analysis}},

language = {english},

publisher = {Cambridge University Press},

isbn_issn = {ISBN-13: 9780521763608},

year = {2010},

edition = {Cambridge},

translation = {0},

length = {250},

url = {http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521763608}

}

**booklet**{RISC4037,author = {E. Kartashova},

title = {{Nonlinear Resonance Analysis}},

language = {english},

publisher = {Cambridge University Press},

isbn_issn = {ISBN-13: 9780521763608},

year = {2010},

edition = {Cambridge},

translation = {0},

length = {250},

url = {http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521763608}

}

### 2009

[Buchberger]

### Automated Reasoning

#### Tudor Jebelean, Bruno Buchberger, Temur Kutsia, Nikolaj Popov, Wolfgang Schreiner, Wolfgang Windsteiger

In: Hagenberg Research, B. Buchberger, M. Affenzeller, A. Ferscha, M. Haller, T. Jebelean, E.P. Klement, P. Paule, G. Pomberger, W. Schreiner, R. Stubenrauch, R. Wagner, G. Weiss, W. Windsteiger (ed.), pp. 63-101. 2009. Springer Dordrecht Heidelberg London New York, ISBN 978-3-642-02126-8. [url]@

author = {Tudor Jebelean and Bruno Buchberger and Temur Kutsia and Nikolaj Popov and Wolfgang Schreiner and Wolfgang Windsteiger},

title = {{Automated Reasoning}},

booktitle = {{Hagenberg Research}},

language = {english},

pages = {63--101},

publisher = {Springer Dordrecht Heidelberg London New York},

isbn_issn = {ISBN 978-3-642-02126-8},

year = {2009},

annote = {2009-00-00-B},

editor = {B. Buchberger and M. Affenzeller and A. Ferscha and M. Haller and T. Jebelean and E.P. Klement and P. Paule and G. Pomberger and W. Schreiner and R. Stubenrauch and R. Wagner and G. Weiss and W. Windsteiger},

refereed = {no},

length = {39},

url = {http://www.springer.com/computer/programming/book/978-3-642-02126-8}

}

**incollection**{RISC3844,author = {Tudor Jebelean and Bruno Buchberger and Temur Kutsia and Nikolaj Popov and Wolfgang Schreiner and Wolfgang Windsteiger},

title = {{Automated Reasoning}},

booktitle = {{Hagenberg Research}},

language = {english},

pages = {63--101},

publisher = {Springer Dordrecht Heidelberg London New York},

isbn_issn = {ISBN 978-3-642-02126-8},

year = {2009},

annote = {2009-00-00-B},

editor = {B. Buchberger and M. Affenzeller and A. Ferscha and M. Haller and T. Jebelean and E.P. Klement and P. Paule and G. Pomberger and W. Schreiner and R. Stubenrauch and R. Wagner and G. Weiss and W. Windsteiger},

refereed = {no},

length = {39},

url = {http://www.springer.com/computer/programming/book/978-3-642-02126-8}

}

[Buchberger]

### Algorithms in Symbolic Computation

#### Peter Paule, Bruno Buchberger, Lena Kartashova, Manuel Kauers, Carsten Schneider, Franz Winkler

In: Hagenberg Research, Bruno Buchberger et al. (ed.), Chapter 1, pp. 5-62. 2009. Springer, 978-3-642-02126-8. [doi] [pdf]@

author = {Peter Paule and Bruno Buchberger and Lena Kartashova and Manuel Kauers and Carsten Schneider and Franz Winkler},

title = {{Algorithms in Symbolic Computation}},

booktitle = {{Hagenberg Research}},

language = {english},

chapter = {1},

pages = {5--62},

publisher = {Springer},

isbn_issn = {978-3-642-02126-8},

year = {2009},

annote = {2009-00-00-C},

editor = {Bruno Buchberger et al.},

refereed = {no},

length = {58},

url = {https://doi.org/10.1007/978-3-642-02127-5_2}

}

**incollection**{RISC3845,author = {Peter Paule and Bruno Buchberger and Lena Kartashova and Manuel Kauers and Carsten Schneider and Franz Winkler},

title = {{Algorithms in Symbolic Computation}},

booktitle = {{Hagenberg Research}},

language = {english},

chapter = {1},

pages = {5--62},

publisher = {Springer},

isbn_issn = {978-3-642-02126-8},

year = {2009},

annote = {2009-00-00-C},

editor = {Bruno Buchberger et al.},

refereed = {no},

length = {58},

url = {https://doi.org/10.1007/978-3-642-02127-5_2}

}

[Kartaschova]

### Dynamics of nonlinear resonances in Hamiltonian systems

#### M.D. Bustamante, E. Kartashova

Europhysics Letters 85, pp. 14004-6. 2009. IOP , 0295-5075 (print) , 1286-4854 (online). [url] [pdf]@

author = {M.D. Bustamante and E. Kartashova},

title = {{Dynamics of nonlinear resonances in Hamiltonian systems}},

language = {english},

abstract = {It is well known that the dynamics of a Hamiltonian system dependscrucially on whether or not it possesses nonlinear resonances. Inthe generic case, the set of nonlinear resonances consists ofindependent clusters of resonantly interacting modes, described by afew low-dimensional dynamical systems. We formulate and prove a newtheorem on integrability which allows us to show that mostfrequently met clusters are described by integrable dynamicalsystems. We argue that construction of clusters can be used as thebase for the Clipping method, substantially more effective for thesesystems than the Galerkin method. The results can be used directlyfor systems with cubic Hamiltonian.},

journal = {Europhysics Letters },

volume = {85},

pages = {14004--6},

publisher = {IOP },

isbn_issn = {0295-5075 (print) , 1286-4854 (online)},

year = {2009},

refereed = {yes},

length = {6},

url = {http://www.iop.org/EJ/abstract/0295-5075/85/1/14004/}

}

**article**{RISC3438,author = {M.D. Bustamante and E. Kartashova},

title = {{Dynamics of nonlinear resonances in Hamiltonian systems}},

language = {english},

abstract = {It is well known that the dynamics of a Hamiltonian system dependscrucially on whether or not it possesses nonlinear resonances. Inthe generic case, the set of nonlinear resonances consists ofindependent clusters of resonantly interacting modes, described by afew low-dimensional dynamical systems. We formulate and prove a newtheorem on integrability which allows us to show that mostfrequently met clusters are described by integrable dynamicalsystems. We argue that construction of clusters can be used as thebase for the Clipping method, substantially more effective for thesesystems than the Galerkin method. The results can be used directlyfor systems with cubic Hamiltonian.},

journal = {Europhysics Letters },

volume = {85},

pages = {14004--6},

publisher = {IOP },

isbn_issn = {0295-5075 (print) , 1286-4854 (online)},

year = {2009},

refereed = {yes},

length = {6},

url = {http://www.iop.org/EJ/abstract/0295-5075/85/1/14004/}

}

[Kartaschova]

### Effect of the dynamical phases on the nonlinear amplitudes' evolution

#### M.D. Bustamante, E. Kartashova

Europhysics Letters 85, pp. 34002-5. 2009. IOP , ISSN (print edition): 0295-5075 ISSN (online): 1286-4854. [url] [pdf]@

author = {M.D. Bustamante and E. Kartashova},

title = {{Effect of the dynamical phases on the nonlinear amplitudes' evolution}},

language = {english},

abstract = {In this Letter we show how the nonlinear evolution of a resonanttriad depends on the special combination of the modes' phases chosenaccordingly to the resonance conditions. This phase combination iscalled dynamical phase, its evolution is studied numerically, bothfor a triad and for a cluster formed by two connected triads. Weshow that dynamical phases, usually regarded as equal to zero orconstants, play substantial role in the dynamics of the clusters.Indeed, effects are (i) to diminish the period of energy exchange$\tau$ within a cluster by 20$\%$ and more; (ii) to diminish, attime scale $\tau$, the variability of wave energies by 25$\%$ andmore; (iii) to generate a new time scale, $T >> \tau$, in which weobserve considerable energy exchange within a cluster, as well as anincreasing in the variability of modes' energies. These findings can be applied, for example, tothe control of energy input, exchange and output in Tokamaks; forexplanation of some experimental results, etc.},

journal = {Europhysics Letters },

volume = {85},

pages = {34002--5},

publisher = {IOP },

isbn_issn = {ISSN (print edition): 0295-5075 ISSN (online): 1286-4854},

year = {2009},

refereed = {yes},

length = {5},

url = {http://www.iop.org/EJ/abstract/0295-5075/85/3/34002}

}

**article**{RISC3788,author = {M.D. Bustamante and E. Kartashova},

title = {{Effect of the dynamical phases on the nonlinear amplitudes' evolution}},

language = {english},

abstract = {In this Letter we show how the nonlinear evolution of a resonanttriad depends on the special combination of the modes' phases chosenaccordingly to the resonance conditions. This phase combination iscalled dynamical phase, its evolution is studied numerically, bothfor a triad and for a cluster formed by two connected triads. Weshow that dynamical phases, usually regarded as equal to zero orconstants, play substantial role in the dynamics of the clusters.Indeed, effects are (i) to diminish the period of energy exchange$\tau$ within a cluster by 20$\%$ and more; (ii) to diminish, attime scale $\tau$, the variability of wave energies by 25$\%$ andmore; (iii) to generate a new time scale, $T >> \tau$, in which weobserve considerable energy exchange within a cluster, as well as anincreasing in the variability of modes' energies. These findings can be applied, for example, tothe control of energy input, exchange and output in Tokamaks; forexplanation of some experimental results, etc.},

journal = {Europhysics Letters },

volume = {85},

pages = {34002--5},

publisher = {IOP },

isbn_issn = {ISSN (print edition): 0295-5075 ISSN (online): 1286-4854},

year = {2009},

refereed = {yes},

length = {5},

url = {http://www.iop.org/EJ/abstract/0295-5075/85/3/34002}

}

[Kartaschova]

### Dynamics of Nonlinear Resonaces

#### E. Kartashova

In: Geophysical research Abstracts, E. Pelinovsky, C. Kharif (ed.), Proceedings of European Geosciences Union, General Assembly 2009, pp. 3232-1. 2009. European Geophysical Union, issn. [url]@

author = {E. Kartashova},

title = {{Dynamics of Nonlinear Resonaces}},

booktitle = {{Geophysical research Abstracts}},

language = {english},

pages = {3232--1},

publisher = {European Geophysical Union},

isbn_issn = {issn},

year = {2009},

editor = {E. Pelinovsky and C. Kharif },

refereed = {yes},

length = {1},

conferencename = {European Geosciences Union, General Assembly 2009},

url = {http://meetingorganizer.copernicus.org/EGU2009/EGU2009-3232-1.pdf}

}

**inproceedings**{RISC3802,author = {E. Kartashova},

title = {{Dynamics of Nonlinear Resonaces}},

booktitle = {{Geophysical research Abstracts}},

language = {english},

pages = {3232--1},

publisher = {European Geophysical Union},

isbn_issn = {issn},

year = {2009},

editor = {E. Pelinovsky and C. Kharif },

refereed = {yes},

length = {1},

conferencename = {European Geosciences Union, General Assembly 2009},

url = {http://meetingorganizer.copernicus.org/EGU2009/EGU2009-3232-1.pdf}

}

[Kartaschova]

### Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

#### A. Constantin, E. Kartashova

EPL 86, pp. 29001-6. 2009. ISSN (print edition): 0295-5075 ISSN (online): 1286-4854. [url] [pdf]@

author = {A. Constantin and E. Kartashova},

title = {{Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves}},

language = {english},

abstract = {The influence of an underlying current on 3-wave interactions ofcapillary water waves is studied. The fact that in irrotational flowresonant 3-wave interactions are not possible can be invalidated bythe presence of an underlying current of constant non-zero vorticity.We show that: 1) wave trains in flows with constant non-zero vorticityare possible only for two-dimensional flows; 2) only positive constant vorticities can trigger theappearance of three-wave resonances; 3) the number of positiveconstant vorticities which do trigger a resonance is countable; 4) themagnitude of a positive constant vorticity triggering a resonance can not be too small.},

journal = {EPL},

volume = {86},

pages = {29001--6},

isbn_issn = {ISSN (print edition): 0295-5075 ISSN (online): 1286-4854},

year = {2009},

refereed = {yes},

length = {6},

url = { http://www.iop.org/EJ/abstract/0295-5075/86/2/29001/}

}

**article**{RISC3812,author = {A. Constantin and E. Kartashova},

title = {{Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves}},

language = {english},

abstract = {The influence of an underlying current on 3-wave interactions ofcapillary water waves is studied. The fact that in irrotational flowresonant 3-wave interactions are not possible can be invalidated bythe presence of an underlying current of constant non-zero vorticity.We show that: 1) wave trains in flows with constant non-zero vorticityare possible only for two-dimensional flows; 2) only positive constant vorticities can trigger theappearance of three-wave resonances; 3) the number of positiveconstant vorticities which do trigger a resonance is countable; 4) themagnitude of a positive constant vorticity triggering a resonance can not be too small.},

journal = {EPL},

volume = {86},

pages = {29001--6},

isbn_issn = {ISSN (print edition): 0295-5075 ISSN (online): 1286-4854},

year = {2009},

refereed = {yes},

length = {6},

url = { http://www.iop.org/EJ/abstract/0295-5075/86/2/29001/}

}

[Kartaschova]

### Nonlinear resonances of water waves

#### E. Kartashova

DCDS Series B 12(3), pp. 607-621. 2009. American Institute of Mathematical Sciences, ISSN 1531-3492 (print) ISSN 1553-524X (electronic). [url] [pdf]@

author = {E. Kartashova},

title = {{Nonlinear resonances of water waves}},

language = {english},

abstract = {In the last fifteen years, a great progress has been made in theunderstanding of the nonlinear resonance dynamics of water waves.Notions of scale- and angle-resonances have been introduced, newtype of energy cascade due to nonlinear resonances in the gravity waterwaves have been discovered, conception of a resonance cluster hasbeen much and successful employed, a novel model of laminated waveturbulence has been developed, etc. etc. Two milestones in this areaof research have to be mentioned: a) development of the $q$-classmethod which is effective for computing integer points on theresonance manifolds, and b) construction of the 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 havebeen found that can be used for explaining numerical andlaboratory results. The aim of this paper is to give a briefoverview of our current knowledge about nonlinear resonances amongwater waves, and formulate three most important open problems at the end.},

journal = {DCDS Series B},

volume = {12},

number = {3},

pages = {607--621},

publisher = {American Institute of Mathematical Sciences},

isbn_issn = {ISSN 1531-3492 (print) ISSN 1553-524X (electronic)},

year = {2009},

refereed = {yes},

length = {15},

url = {http://aimsciences.org/journals/displayPapers1.jsp?pubID=317}

}

**article**{RISC3822,author = {E. Kartashova},

title = {{Nonlinear resonances of water waves}},

language = {english},

abstract = {In the last fifteen years, a great progress has been made in theunderstanding of the nonlinear resonance dynamics of water waves.Notions of scale- and angle-resonances have been introduced, newtype of energy cascade due to nonlinear resonances in the gravity waterwaves have been discovered, conception of a resonance cluster hasbeen much and successful employed, a novel model of laminated waveturbulence has been developed, etc. etc. Two milestones in this areaof research have to be mentioned: a) development of the $q$-classmethod which is effective for computing integer points on theresonance manifolds, and b) construction of the 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 havebeen found that can be used for explaining numerical andlaboratory results. The aim of this paper is to give a briefoverview of our current knowledge about nonlinear resonances amongwater waves, and formulate three most important open problems at the end.},

journal = {DCDS Series B},

volume = {12},

number = {3},

pages = {607--621},

publisher = {American Institute of Mathematical Sciences},

isbn_issn = {ISSN 1531-3492 (print) ISSN 1553-524X (electronic)},

year = {2009},

refereed = {yes},

length = {15},

url = {http://aimsciences.org/journals/displayPapers1.jsp?pubID=317}

}

[Kartaschova]

### Austrian-Ukrainian Project CENREC as example of information support of activity of international scientific community

#### E. Kartashova, M. Lvov

Informational technologies in education 1(3), pp. 57-63. 2009. Kherson State University, 1998-6939. [url]@

author = {E. Kartashova and M. Lvov},

title = {{Austrian-Ukrainian Project CENREC as example of information support of activity of international scientific community}},

language = {english},

journal = {Informational technologies in education},

volume = {1},

number = {3},

pages = {57--63},

publisher = {Kherson State University},

isbn_issn = {1998-6939},

year = {2009},

refereed = {yes},

length = {7},

url = {http://ite.ksu.ks.ua}

}

**article**{RISC3834,author = {E. Kartashova and M. Lvov},

title = {{Austrian-Ukrainian Project CENREC as example of information support of activity of international scientific community}},

language = {english},

journal = {Informational technologies in education},

volume = {1},

number = {3},

pages = {57--63},

publisher = {Kherson State University},

isbn_issn = {1998-6939},

year = {2009},

refereed = {yes},

length = {7},

url = {http://ite.ksu.ks.ua}

}

[Kartaschova]

### Discrete Wave Turbulence

#### E. Kartashova

Europhysics Letters 87, pp. 44001-p5. 2009. 0295-5075 (print) , 1286-4854 (online). EDITOR CHOICE. [url] [pdf]@

author = {E. Kartashova},

title = {{Discrete Wave Turbulence}},

language = {english},

abstract = {In this Letter we propose discrete wave turbulence (DWT) as a counterpart of classical statistical wave turbulence (SWT). DWT is characterized by resonance clustering, not by the size of clusters, i.e. it includes, but is not reduced to, the study of low-dimensional systems. Clusters with integrable and chaotic dynamics co-exist in different sub-spaces of the $\mathbf{k}$-space. NR-diagrams are introduced, a graphical representation of an arbitrary resonance cluster allowing to reconstruct uniquely dynamical system describing the cluster. DWT is shown to be a novel research field in nonlinear science, with its own methods, achievements and application areas.},

journal = {Europhysics Letters},

volume = {87},

pages = {44001--p5},

isbn_issn = { 0295-5075 (print) , 1286-4854 (online)},

year = {2009},

note = {EDITOR CHOICE},

refereed = {yes},

length = {5},

url = {http://www.iop.org/EJ/abstract/0295-5075/87/4/44001}

}

**article**{RISC3873,author = {E. Kartashova},

title = {{Discrete Wave Turbulence}},

language = {english},

abstract = {In this Letter we propose discrete wave turbulence (DWT) as a counterpart of classical statistical wave turbulence (SWT). DWT is characterized by resonance clustering, not by the size of clusters, i.e. it includes, but is not reduced to, the study of low-dimensional systems. Clusters with integrable and chaotic dynamics co-exist in different sub-spaces of the $\mathbf{k}$-space. NR-diagrams are introduced, a graphical representation of an arbitrary resonance cluster allowing to reconstruct uniquely dynamical system describing the cluster. DWT is shown to be a novel research field in nonlinear science, with its own methods, achievements and application areas.},

journal = {Europhysics Letters},

volume = {87},

pages = {44001--p5},

isbn_issn = { 0295-5075 (print) , 1286-4854 (online)},

year = {2009},

note = {EDITOR CHOICE},

refereed = {yes},

length = {5},

url = {http://www.iop.org/EJ/abstract/0295-5075/87/4/44001}

}

### 2008

[Kartaschova]

### Kinematic Theory of Nonlinear Resonances

#### Elena Kartashova

10, Geophysical Research Abstracts edition, 2008. EGU (European Geophysical Union), isbn. [pdf]@

author = {Elena Kartashova},

title = {{Kinematic Theory of Nonlinear Resonances}},

language = {english},

volume = {10},

publisher = { EGU (European Geophysical Union)},

isbn_issn = {isbn},

year = {2008},

edition = {Geophysical Research Abstracts},

translation = {0},

length = {1}

}

**booklet**{RISC3388,author = {Elena Kartashova},

title = {{Kinematic Theory of Nonlinear Resonances}},

language = {english},

volume = {10},

publisher = { EGU (European Geophysical Union)},

isbn_issn = {isbn},

year = {2008},

edition = {Geophysical Research Abstracts},

translation = {0},

length = {1}

}

[Kartaschova]

### Cluster Dynamics of Planetary Waves

#### E. Kartashova, V. L`vov

Europhys. Letters 83, pp. 50012-6. 2008. ISSN: 0295-5075 (Print) 1286-4854 (Online). [url] [pdf]@

author = {E. Kartashova and V. L`vov},

title = {{Cluster Dynamics of Planetary Waves}},

language = {english},

abstract = {Dynamics of nonlinear atmospheric planetary waves is determined by asmall number of independent wave clusters consisting of a fewconnected resonant triads. We classified different types of connections betweenneighbor triads that determines general dynamics of acluster. Each connection type corresponds to substantially different scenarios ofenergy flux among the modes. The general approach can be applieddirectly to various mesoscopic systems with 3-mode interactions, encountered inhydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.},

journal = {Europhys. Letters},

volume = {83},

pages = {50012--6},

isbn_issn = {ISSN: 0295-5075 (Print) 1286-4854 (Online)},

year = {2008},

refereed = {yes},

length = {6},

url = {http://arxiv.org/abs/0801.3374}

}

**article**{RISC3393,author = {E. Kartashova and V. L`vov},

title = {{Cluster Dynamics of Planetary Waves}},

language = {english},

abstract = {Dynamics of nonlinear atmospheric planetary waves is determined by asmall number of independent wave clusters consisting of a fewconnected resonant triads. We classified different types of connections betweenneighbor triads that determines general dynamics of acluster. Each connection type corresponds to substantially different scenarios ofenergy flux among the modes. The general approach can be applieddirectly to various mesoscopic systems with 3-mode interactions, encountered inhydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.},

journal = {Europhys. Letters},

volume = {83},

pages = {50012--6},

isbn_issn = {ISSN: 0295-5075 (Print) 1286-4854 (Online)},

year = {2008},

refereed = {yes},

length = {6},

url = {http://arxiv.org/abs/0801.3374}

}

[Kartaschova]

### Resonant interactions of nonlinear water waves in a finite basin

#### E. Kartashova, S. Nazarenko, O. Rudenko

Physical Review E 78 (016304), pp. 1-9. 2008. American Physical Society, 1539-3755 (print) , 1550-2376 (online). [url] [pdf]@

author = {E. Kartashova and S. Nazarenko and O. Rudenko},

title = {{Resonant interactions of nonlinear water waves in a finite basin}},

language = {english},

abstract = {We study exact four-wave resonances among gravity water waves in asquare box with periodic boundary conditions. We show that theseresonant quartets are linked with each other by shared Fourier modesin such a way that they form independent clusters. These clusterscan be formed by two types of quartets: (1) {\it angle-resonances} which cannot directly cascade energy but which canredistribute it among the initially excited modes and (2) {\itscale-resonances} which are much more rare but which are the onlyones that can transfer energy between different scales. We find suchresonant quartets and their clusters numerically on the set of 1000x 1000 modes, classify and quantify them and discuss consequences ofthe obtained cluster structure for the wavefield evolution. Finitebox effects and associated resonant interaction among discrete wavemodes appear to be important in most numerical and laboratoryexperiments on the deep water gravity waves, and our work is aimedat aiding the interpretation of the experimental and numerical data.},

journal = {Physical Review E},

volume = {78 },

number = {016304},

pages = {1--9},

publisher = {American Physical Society},

isbn_issn = {1539-3755 (print) , 1550-2376 (online)},

year = {2008},

refereed = {yes},

length = {9},

url = {http://link.aps.org/abstract/PRE/v78/e016304}

}

**article**{RISC3410,author = {E. Kartashova and S. Nazarenko and O. Rudenko},

title = {{Resonant interactions of nonlinear water waves in a finite basin}},

language = {english},

abstract = {We study exact four-wave resonances among gravity water waves in asquare box with periodic boundary conditions. We show that theseresonant quartets are linked with each other by shared Fourier modesin such a way that they form independent clusters. These clusterscan be formed by two types of quartets: (1) {\it angle-resonances} which cannot directly cascade energy but which canredistribute it among the initially excited modes and (2) {\itscale-resonances} which are much more rare but which are the onlyones that can transfer energy between different scales. We find suchresonant quartets and their clusters numerically on the set of 1000x 1000 modes, classify and quantify them and discuss consequences ofthe obtained cluster structure for the wavefield evolution. Finitebox effects and associated resonant interaction among discrete wavemodes appear to be important in most numerical and laboratoryexperiments on the deep water gravity waves, and our work is aimedat aiding the interpretation of the experimental and numerical data.},

journal = {Physical Review E},

volume = {78 },

number = {016304},

pages = {1--9},

publisher = {American Physical Society},

isbn_issn = {1539-3755 (print) , 1550-2376 (online)},

year = {2008},

refereed = {yes},

length = {9},

url = {http://link.aps.org/abstract/PRE/v78/e016304}

}

[Raab]

### Symbolic Computations for Nonlinear Wave Resonances

#### E. Kartashova, C. Raab, Ch. Feurer, G. Mayrhofer, W. Schreiner

In: "Extreme Ocean Waves", E. Pelinovsky, Ch. Kharif (ed.), pp. 97-128. 2008. Springer, ISBN: 978-1-4020-8313-6. [url] [pdf]@

author = {E. Kartashova and C. Raab and Ch. Feurer and G. Mayrhofer and W. Schreiner},

title = {{Symbolic Computations for Nonlinear Wave Resonances}},

booktitle = {{"Extreme Ocean Waves"}},

language = {english},

abstract = {Nonlinear dynamics and pattern formation in the systems with quadratic nonlinearity is computed symbolically by specially developed MATHEMATICA package. A Web interface for the presented methods is developed, which turns the implementations from only locally available software to Web-based services that can be accessed from any computer in the Internet that is equipped with a Web browser. In particular, the results are not bound to the current Mathematica implementation but can be adapted to any other computer algebra system (e.g. Maple) or numerical software system (e.g.MATLAB) of similar expressiveness. Barotropic vorticity equation (=Hasegawa-Mima equation) with zero boundary conditions on a square is taken as a main example.},

pages = {97--128},

publisher = {Springer},

isbn_issn = {ISBN: 978-1-4020-8313-6},

year = {2008},

editor = {E. Pelinovsky and Ch. Kharif},

refereed = {yes},

length = {31},

url = {http://www.springer.com/geosciences/oceanography/book/978-1-4020-8313-6}

}

**incollection**{RISC3344,author = {E. Kartashova and C. Raab and Ch. Feurer and G. Mayrhofer and W. Schreiner},

title = {{Symbolic Computations for Nonlinear Wave Resonances}},

booktitle = {{"Extreme Ocean Waves"}},

language = {english},

abstract = {Nonlinear dynamics and pattern formation in the systems with quadratic nonlinearity is computed symbolically by specially developed MATHEMATICA package. A Web interface for the presented methods is developed, which turns the implementations from only locally available software to Web-based services that can be accessed from any computer in the Internet that is equipped with a Web browser. In particular, the results are not bound to the current Mathematica implementation but can be adapted to any other computer algebra system (e.g. Maple) or numerical software system (e.g.MATLAB) of similar expressiveness. Barotropic vorticity equation (=Hasegawa-Mima equation) with zero boundary conditions on a square is taken as a main example.},

pages = {97--128},

publisher = {Springer},

isbn_issn = {ISBN: 978-1-4020-8313-6},

year = {2008},

editor = {E. Pelinovsky and Ch. Kharif},

refereed = {yes},

length = {31},

url = {http://www.springer.com/geosciences/oceanography/book/978-1-4020-8313-6}

}