# CENREC [CENREC]

### Project Duration

01/01/2009 - 31/12/2010

Project Website

## Publications

### Discrete wave turbulence of rotational capillary water waves

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

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

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

#### E. Kartashova

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

### 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]
@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}
}

### Turbulence of capillary waves revisited

#### E. Kartashova, A. Kartashov

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

### Nonlinear Resonance Analysis

#### E. Kartashova

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

### Algorithms in Symbolic Computation

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

In: Hagenberg Research, , Chapter 1, pp. 5-62. 2009. Springer, 978-3-642-02126-8. [pdf]
@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}
}

### 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]
@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/}
}

### 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]
@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}
}

### 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]
@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}
}

### Discrete Wave Turbulence

#### E. Kartashova

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