Resonance clustering in wave turbulent regimes: Integrable dynamics Elena Kartashova†, Miguel D. Bustamante∗ †RISC, J.Kepler University, Linz 4040, Austria ∗School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland 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 cluster- ing. 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 paper we study 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 establish that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others – only for par- ticular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed. PACS numbers: 47.10.Df, 47.10.Fg, 02.70.Dh Contents I. INTRODUCTION I. Introduction 1 The broad structure of modern nonlinear science born at the edge of physics and mathematics includes an enor- II. NR-diagrams 3 mous number of applications in cosmology, biochemistry, electronics, optics, hydrodynamics, economics, neuro- III. Dynamical invariants 5 science, etc. The emergence of nonlinear science itself as A. Definition 5 a collective interdisciplinary activity is due to the aware- B. Example: damped harmonic oscillator 5 ness that its dynamic concepts first observed and un- derstood in one field (for example, population biology, 1. Dynamical invariants, CLs and solutions 5 flame-front propagation, non-linear optics or planetary 2. Description of a laboratory experiment 7 motion) could be useful in others (such as in chemical 3. Constructionofconservationlaw 8 dynamics, neuroscience, plasma confinement or weather prediction). The theory of integrable Hamiltonian sys- IV. Triad 8 tems, a generalization of the classical theory of differ- A. Integrability 8 ential equations, is the nucleus of the whole nonlinear B. Amplitude-phase representation 9 science. Various classifications of integrable systems are C. Solutions for amplitudes 9 presently known which turned out to be quite useful for D. Solution for dynamical phase 9 physical applications. Classifications are known based on E. Dynamical invariant 10 the various intrinsic properties of integrable systems [53]: symmetries, conservation laws, Lax-pairs, etc. In [3] the F. Special case H =0 11 T general classification of integrable Hamiltonian systems arXiv:1002.4994v1 [nlin.SI] 26 Feb 2010 is presented based on the form of their topological invari- V. Generic clusters 11 ants. The usefulness of this classification is demonstrated A. Kite 11 in several problems on solid mechanics. In particular, it B. Butterfly 11 is proven that two famous problems – the Euler case in C. Ray 13 rigid body dynamics and the Jacobi problem of geodesics D. Star 14 on the ellipsoid– are orbitally equivalent. In [12] the idea of classification is presented based on normal forms of VI. Coupling coefficient 15 a certain class of bi-hamiltonian PDEs. Miscellaneous hierarchies of integrable PDEs are presented in [48]. VII. Summary 16 The list can be prolonged further but the main point for us presently is the following: the notion of integrabil- References 17 ity itself is ambitious! There are many quite different def- initions of integrability, for instance integrability in terms of elementary functions (equationy ¨ = y has the explicit solution y = a sin (x + b) ); integrability− modulo class of ∗Electronic address: [email protected], [email protected] (equationy ¨ = f(y) has general solutions in 2 FIG. 1: Color online. Schematic representation of wave turbulent regimes. terms of elliptic functions), etc. An example of less obvi- terplay generates three possible wave turbulent regimes: ous definition of integrability is C-integrability, first in- kinetic, discrete and mesoscopic as it is shown in Fig. 1 troduced in [6]: integrability modulo change of variables, (see [10] for more discussion). meaning that a nonlinear equation is called C-integrable The existence of mesoscopic regime has been first con- if it can be turned into a linear equation by an appropri- firmed in numerical simulations with dynamical equa- ate invertible change of variables. For instance, Thomas tions for surface gravity waves in [67], while discrete equation ψxy + αψx + βψy + ψxψy = 0 is C-integrable. regime has been first described in [25]. These theoret- Profound discussion on the subject can be found in [34]. ical findings are confirmed by numerous laboratory ex- In the present paper, integrability is interpreted in terms periments. For instance, in the experiments with gravity of the existence of a number of independent dynamical surface wave turbulence in a laboratory flume, [11], only invariants of the system; for each in-this-sense-integrable a discrete regime has been identified while in [62] coex- system, solutions are then written out in quadratures. istence of both types of time evolution has been estab- The dynamical systems we are interested in, describe lished. Taking into account additional physical param- nonlinear resonance clusters appearing in evolutionary eters in a wave system transition from kinetic to meso- dispersive PDEs in two space variables. Nonlinear reso- scopic regime can be observed as it was demonstrated in nances are ubiquitous in physics. They appear in a great [9] for capillary water waves, with and without rotation. amount of typical mechanical systems [13, 41], in engi- From a mathematical point of view, the very special neering [8, 17, 42, 57], astronomy [40], biology [15], etc. role of resonant solutions has been first demonstrated by etc. Euler equations, regarded with various boundary Poincar´ewho proved, using Calogero’s terminology, that conditions and specific values of some parameters, de- a nonlinear ODE is C-integrable if it has no resonance so- scribe an enormous number of nonlinear dispersive wave lutions (see [2] and refs. therein). This statement allows systems (capillary waves, surface water waves, atmo- the following Hamiltonian formulation [63]: spheric planetary waves, drift waves in plasma, etc.) all ∗ i a˙ k = ∂ /∂ak, (1) possessing nonlinear resonances. H The classical approach of statistical wave turbulence where ak is the amplitude of the Fourier mode corre- theory in a nonlinear wave system assumes weak non- sponding to the wavevector k and the Hamiltonian is linearity, randomness of phases, infinite-box limit, and represented as an expansion in powers which areH pro- Hj existence of an inertial interval in wavenumber space portional to the product of j amplitudes ak. Then the (k0, k1), where energy input and dissipation are bal- cubic Hamiltonian has the form anced. Under these assumptions, the wave system is energy conserving, and wave kinetic equations describ- = V 3 a∗a a δ1 + complex conj., H3 12 1 2 3 23 ing the wave spectrum have stationary solutions in the k1,k2,k3 form of Kolmogorov-Zakharov (KZ) energy power spec- X −α k tra k ,α> 0, ([55, 63, 66], etc.). where for brevity we introduced the notation aj a j 1 ≡ As it was first established in the frame of the model and δ23 δ(k3 k1 k2) is the Kronecker symbol. If = 0,≡ three-wave− resonant− processes are dominant. of laminated turbulence, [22], KZ-spectra have “gaps” H3 6 formed by exact and quasi-resonances (that is, resonances These satisfy the resonance conditions: with small enough resonance broadening). This yields ω(k )+ ω(k ) ω(k )=0 two distinct layers of turbulence in an arbitrary nonlin- 1 2 − 3 (2) ear wave system – continuous and discrete – and their in- k1 + k2 k3 =0, ( − 3 where ω(k) is a dispersion relation for the linear wave H18L H1L frequency. Further on, the notation ωk is used for ω(k). The corresponding dynamical system has a general form H4L H1L ˙ k k 1 ∗ ∗ 1 iBk = V12B1B2δ12 +2Vk2 B1B2 δk3) (3) H1L k1,k2 X (notations Bj are used further on for the resonant H2L H1L modes). If = 0, four-wave resonances have to be H3 studied, and so on. To confirm that 3 = 0 and three- wave resonances are dominant, one hasH to6 find solutions 3 of (2) and check that V12 = 0 at least at some resonant triads. Afterwards the corresponding6 dynamical system FIG. 2: Topological structure of the cluster set for the oceanic planetary waves, ω 1/√m2 + n2, in the domain m, n 50. has to be studied. 7 types of resonance∼ clusters have been found, the number≤ of It has been first proven in [21] that for a big class the clusters of each type is shown in parenthesis. of physically relevant dispersion functions ω, the set of all wavevectors satisfying (2) can be divided into non- intersecting classes and solutions of (2) can be looked and is known to be integrable (e.g. [61]), with two con- for in each class separately. The method of q-class de- servation laws in the Manley–Rowe form being composition first introduced in [23] has been developed I = B 2 + B 2, I = B 2 + B 2. specially for solving systems of the form (2) in integers; 23 | 2| | 3| 13 | 1| | 3| details of its implementation for various rational and irra- Due to the criterion of nonlinear instability for a triad tional dispersion functions are given in [27]–[29]. General [16], the mode with maximal frequency, ω , is unstable description of the q-class method and corresponding pro- 3 while the modes ω and ω are neutral. Originally, this gramming codes are given in [26], in Ch.3 and Appendix 1 2 fact has been deduced directly from the equations of mo- correspondingly.