The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance...The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants(gMRC)and give a constructive method how to compute them.We also give details of a Mathematica implementation of this method.WhileMRC provide integrability of the underlying dynamical system,gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.展开更多
Two fundamental facts of the modern wave turbulence theory are 1)existence of power energy spectra in k-space,and 2)existence of“gaps”in this spectra corresponding to the resonance clustering.Accordingly,three wave ...Two fundamental facts of the modern wave turbulence theory are 1)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:kinetic,described by wave kinetic equations and power energy spectra;discrete,characterized by resonance clustering;and mesoscopic,where both types of wave field time evolution coexist.In this review paper we present the results on integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes.Using a novel method based on the notion of dynamical invariant we show that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others-only for particular initial conditions.We also identify chaotic behaviour in some cases.Physical implications of the results obtained are discussed.展开更多
The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10^(12) and more.The equations...The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10^(12) and more.The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion.It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary:(1)one-class-case algorithm for waves interacting through scales,and(2)two-class-case algorithm for waves interacting through phases.In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-classcase generic algorithm.展开更多
基金support of the Austrian Science Foundation(FWF)under project P22943-N18”Nonlinear resonances of water waves”and in part–by the Project of Knowledge Innovation Program(PKIP)of Chinese Academy of Sciences,Grant No.KJCX2.YW.W10.E.K.is very much obliged to the organizing committee of the program”New Directions in Turbulence”(KITPC/ITP-CAS,2012)and the hospitality of Kavli ITP,Beijing,where part of this work has been accomplished.
文摘The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants(gMRC)and give a constructive method how to compute them.We also give details of a Mathematica implementation of this method.WhileMRC provide integrability of the underlying dynamical system,gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.
基金funded by European Commission Framework 6 Programme for Integrated Infrastructures Initiatives under the project SCIEnce(Contract No.026133)E.Kartashova acknowledges the support of the Austrian Science Foundation(FWF)under projects P20164 and P22943.
文摘Two fundamental facts of the modern wave turbulence theory are 1)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:kinetic,described by wave kinetic equations and power energy spectra;discrete,characterized by resonance clustering;and mesoscopic,where both types of wave field time evolution coexist.In this review paper we present the results on integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes.Using a novel method based on the notion of dynamical invariant we show that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others-only for particular initial conditions.We also identify chaotic behaviour in some cases.Physical implications of the results obtained are discussed.
文摘The model of laminated wave turbulence puts forth a novel computational problem–construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order 10^(12) and more.The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion.It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary:(1)one-class-case algorithm for waves interacting through scales,and(2)two-class-case algorithm for waves interacting through phases.In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-classcase generic algorithm.