# Symbolic Differential Computation [SFB F1304-3]

### Project Lead

### Project Duration

01/04/2004 - 30/09/2008### Project URL

Project Website## Publications

### 2018

### Rational General Solutions of Systems of First-Order Partial Differential Equations

#### Georg Grasegger, Alberto Lastra, J. Rafael Sendra, Franz Winkler

Journal of Computational and Applied Mathematics 331, pp. 88-103. 2018. ISSN: 0377-0427.@

author = {Georg Grasegger and Alberto Lastra and J. Rafael Sendra and Franz Winkler},

title = {{Rational General Solutions of Systems of First-Order Partial Differential Equations}},

language = {english},

journal = {Journal of Computational and Applied Mathematics},

volume = {331},

pages = {88--103},

isbn_issn = {ISSN: 0377-0427},

year = {2018},

refereed = {yes},

length = {16}

}

**article**{RISC5509,author = {Georg Grasegger and Alberto Lastra and J. Rafael Sendra and Franz Winkler},

title = {{Rational General Solutions of Systems of First-Order Partial Differential Equations}},

language = {english},

journal = {Journal of Computational and Applied Mathematics},

volume = {331},

pages = {88--103},

isbn_issn = {ISSN: 0377-0427},

year = {2018},

refereed = {yes},

length = {16}

}

### 2016

### A decision algorithm for rational general solutions of first-order algebraic ODEs

#### G. Grasegger, N.T. Vo, F. Winkler

In: Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016), Universidad de la Rioja, J. Heras and A. Romero (eds.) (ed.), pp. 101-104. 2016. 978-84-608-9024-9.@

author = {G. Grasegger and N.T. Vo and F. Winkler},

title = {{A decision algorithm for rational general solutions of first-order algebraic ODEs}},

booktitle = {{Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016)}},

language = {english},

pages = {101--104},

isbn_issn = {978-84-608-9024-9},

year = {2016},

editor = {Universidad de la Rioja and J. Heras and A. Romero (eds.)},

refereed = {yes},

length = {4}

}

**inproceedings**{RISC5400,author = {G. Grasegger and N.T. Vo and F. Winkler},

title = {{A decision algorithm for rational general solutions of first-order algebraic ODEs}},

booktitle = {{Proceedings XV Encuentro de Algebra Computacional y Aplicaciones (EACA 2016)}},

language = {english},

pages = {101--104},

isbn_issn = {978-84-608-9024-9},

year = {2016},

editor = {Universidad de la Rioja and J. Heras and A. Romero (eds.)},

refereed = {yes},

length = {4}

}

### 2015

### Computation of Dimension in Filtered Free Modules by Gröbner Reduction

#### Christoph Fuerst, Guenter Landsmann

In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, ACM (ed.), Proceedings of ISSAC '15, pp. 181-188. 2015. 978-1-4503-3435-8. [url]@

author = {Christoph Fuerst and Guenter Landsmann},

title = {{Computation of Dimension in Filtered Free Modules by Gröbner Reduction}},

booktitle = {{Proceedings of the International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {181--188},

isbn_issn = {978-1-4503-3435-8},

year = {2015},

editor = {ACM},

refereed = {yes},

length = {8},

conferencename = {ISSAC '15},

url = {http://doi.acm.org/10.1145/2755996.2756680}

}

**inproceedings**{RISC5154,author = {Christoph Fuerst and Guenter Landsmann},

title = {{Computation of Dimension in Filtered Free Modules by Gröbner Reduction}},

booktitle = {{Proceedings of the International Symposium on Symbolic and Algebraic Computation}},

language = {english},

pages = {181--188},

isbn_issn = {978-1-4503-3435-8},

year = {2015},

editor = {ACM},

refereed = {yes},

length = {8},

conferencename = {ISSAC '15},

url = {http://doi.acm.org/10.1145/2755996.2756680}

}

### Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one

#### A. Lastra, J.R. Sendra, L.X.C. Ngô, F. Winkler

Publ.Math.Debrecen(86/1-2), pp. 49-69. 2015. 0033-3883.@

author = {A. Lastra and J.R. Sendra and L.X.C. Ngô and F. Winkler},

title = {{Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one}},

language = {english},

journal = {Publ.Math.Debrecen},

number = {86/1-2},

pages = {49--69},

isbn_issn = {0033-3883},

year = {2015},

refereed = {yes},

length = {21}

}

**article**{RISC5204,author = {A. Lastra and J.R. Sendra and L.X.C. Ngô and F. Winkler},

title = {{Rational general solutions of systems of autonomous ordinary differential equations of algebro-geometric dimension one}},

language = {english},

journal = {Publ.Math.Debrecen},

number = {86/1-2},

pages = {49--69},

isbn_issn = {0033-3883},

year = {2015},

refereed = {yes},

length = {21}

}

### Birational transformations preserving rational solutions of algebraic ordinary differential equations

#### L.X.C. Ngô, J.R. Sendra, F. Winkler

J. Computational and Applied Mathematics(286), pp. 114-127. 2015. 0377-0427.@

author = {L.X.C. Ngô and J.R. Sendra and F. Winkler},

title = {{Birational transformations preserving rational solutions of algebraic ordinary differential equations}},

language = {english},

journal = {J. Computational and Applied Mathematics},

number = {286},

pages = {114--127},

isbn_issn = {0377-0427},

year = {2015},

refereed = {yes},

length = {14}

}

**article**{RISC5205,author = {L.X.C. Ngô and J.R. Sendra and F. Winkler},

title = {{Birational transformations preserving rational solutions of algebraic ordinary differential equations}},

language = {english},

journal = {J. Computational and Applied Mathematics},

number = {286},

pages = {114--127},

isbn_issn = {0377-0427},

year = {2015},

refereed = {yes},

length = {14}

}

### Algebraic General Solutions of First Order Algebraic ODEs

#### N. T. Vo, F. Winkler

In: Computer Algebra in Scientific Computing, Vladimir P. Gerdt et. al. (ed.), Lecture Notes in Computer Science 9301, pp. 479-492. 2015. Springer International Publishing, ISSN 0302-9743. [url]@

author = {N. T. Vo and F. Winkler},

title = {{Algebraic General Solutions of First Order Algebraic ODEs}},

booktitle = {{Computer Algebra in Scientific Computing}},

language = {english},

abstract = {In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.},

series = {Lecture Notes in Computer Science},

volume = {9301},

pages = {479--492},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {Vladimir P. Gerdt et. al.},

refereed = {yes},

length = {14},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-24021-3_35.pdf}

}

**inproceedings**{RISC5194,author = {N. T. Vo and F. Winkler},

title = {{Algebraic General Solutions of First Order Algebraic ODEs}},

booktitle = {{Computer Algebra in Scientific Computing}},

language = {english},

abstract = {In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.},

series = {Lecture Notes in Computer Science},

volume = {9301},

pages = {479--492},

publisher = {Springer International Publishing},

isbn_issn = {ISSN 0302-9743},

year = {2015},

editor = {Vladimir P. Gerdt et. al.},

refereed = {yes},

length = {14},

url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-24021-3_35.pdf}

}

### 2013

### Rational general solutions of higher order algebraic ODEs

#### Y. Huang, L.X.C. Ngo, F. Winkler

J. Systems Science and Complexity (JSSC) 26/2, pp. 261-280. 2013. 1009-6124.@

author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of higher order algebraic ODEs}},

language = {english},

journal = {J. Systems Science and Complexity (JSSC)},

volume = {26/2},

pages = {261--280},

isbn_issn = {1009-6124},

year = {2013},

refereed = {yes},

length = {20}

}

**article**{RISC4640,author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of higher order algebraic ODEs}},

language = {english},

journal = {J. Systems Science and Complexity (JSSC)},

volume = {26/2},

pages = {261--280},

isbn_issn = {1009-6124},

year = {2013},

refereed = {yes},

length = {20}

}

### Rational general solutions of trivariate rational systems of autonomous ODEs

#### Y. Huang, L.X.C. Ngo, F. Winkler

Mathematics in Computer Science 6/4, pp. 361-374. 2013. 1661-8270.@

author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of trivariate rational systems of autonomous ODEs}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {6/4},

pages = {361--374},

isbn_issn = {1661-8270},

year = {2013},

refereed = {yes},

length = {14}

}

**article**{RISC4641,author = {Y. Huang and L.X.C. Ngo and F. Winkler},

title = {{Rational general solutions of trivariate rational systems of autonomous ODEs}},

language = {english},

journal = {Mathematics in Computer Science},

volume = {6/4},

pages = {361--374},

isbn_issn = {1661-8270},

year = {2013},

refereed = {yes},

length = {14}

}

### 2012

### Computer algebra methods for pattern recognition: systems with complex order

#### F. Winkler, M. Hudayberdiev, G. Judakova

In: Proceedings INTELS 2012 (Moscow), - (ed.), Proceedings of INTELS 2012, pp. 148-150. 2012. 978-5-93347-432-6.@

author = {F. Winkler and M. Hudayberdiev and G. Judakova},

title = {{Computer algebra methods for pattern recognition: systems with complex order}},

booktitle = {{Proceedings INTELS 2012 (Moscow)}},

language = {english},

pages = {148--150},

isbn_issn = {978-5-93347-432-6},

year = {2012},

editor = {-},

refereed = {yes},

length = {3},

conferencename = {INTELS 2012}

}

**inproceedings**{RISC4639,author = {F. Winkler and M. Hudayberdiev and G. Judakova},

title = {{Computer algebra methods for pattern recognition: systems with complex order}},

booktitle = {{Proceedings INTELS 2012 (Moscow)}},

language = {english},

pages = {148--150},

isbn_issn = {978-5-93347-432-6},

year = {2012},

editor = {-},

refereed = {yes},

length = {3},

conferencename = {INTELS 2012}

}

### Classification of algebraic ODEs with respect to rational solvability

#### L.X.C. Ngo, J.R. Sendra, F. Winkler

Computational Algebraic and Analytic Geometry, Contemporary Mathematics(572), pp. 193-210. 2012. AMS, 0271-4132.@

author = {L.X.C. Ngo and J.R. Sendra and F. Winkler},

title = {{Classification of algebraic ODEs with respect to rational solvability}},

language = {english},

journal = {Computational Algebraic and Analytic Geometry, Contemporary Mathematics},

number = {572},

pages = {193--210},

publisher = {AMS},

isbn_issn = {0271-4132},

year = {2012},

refereed = {yes},

length = {18}

}

**article**{RISC4637,author = {L.X.C. Ngo and J.R. Sendra and F. Winkler},

title = {{Classification of algebraic ODEs with respect to rational solvability}},

language = {english},

journal = {Computational Algebraic and Analytic Geometry, Contemporary Mathematics},

number = {572},

pages = {193--210},

publisher = {AMS},

isbn_issn = {0271-4132},

year = {2012},

refereed = {yes},

length = {18}

}

### The role of Symbolic Computation in Mathematics

#### F. Winkler

In: Proceedings XIII Encuentro de Algebra Computacional y Aplicaciones (EACA 2012), J.R. Sendra and C. Villarino (ed.), pp. 33-34. 2012. 978-84-8138-770-4.@

author = {F. Winkler},

title = {{The role of Symbolic Computation in Mathematics}},

booktitle = {{Proceedings XIII Encuentro de Algebra Computacional y Aplicaciones (EACA 2012)}},

language = {english},

pages = {33--34},

isbn_issn = {978-84-8138-770-4},

year = {2012},

editor = {J.R. Sendra and C. Villarino},

refereed = {yes},

length = {2}

}

**inproceedings**{RISC4638,author = {F. Winkler},

title = {{The role of Symbolic Computation in Mathematics}},

booktitle = {{Proceedings XIII Encuentro de Algebra Computacional y Aplicaciones (EACA 2012)}},

language = {english},

pages = {33--34},

isbn_issn = {978-84-8138-770-4},

year = {2012},

editor = {J.R. Sendra and C. Villarino},

refereed = {yes},

length = {2}

}

### 2010

### 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}

}

### 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),}

}

### 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}

}

### 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}

}

### 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

### 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. [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}

}

**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}

}

### 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/}

}

### 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}

}

### 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}

}