OFFPRINT Discrete wave turbulence

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 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. Copyright c © EPLA, 2009 Introduction. – In the years between 1960 and 1990 a great volume of research has been performed in the area of wave interactions [1], a comprehensive account of theory and experiment can be found in [2]. The notion of a wave kinetic equation was introduced [3]. Using some statistical assumptions, general methods for deriving kinetic equations and their stationary solutions (energy spectra) were developed and statistical wave turbulence (SWT) theory was founded [4], with finite-size effects left aside. Their preliminary studies were performed in [5,6], where it was established that nonlinear resonances are divided into dynamically independent, non-intersecting clusters. Explicit constructing of resonance clustering became a challenge of great intricacy, because no analytical methods for solving resonance conditions were known (the problem in its general form is equivalent to Hilbert’s 10th problem [7]). Brute-force computer computations do not help either, while integers to be dealt with are too big. The problem has been recently solved [8]. For a complete resonance set, classical resonance curves [9] are not anymore a suitable representation. Instead, representation by a hyper-graph on a plane was introduced in [10] (for 3-wave systems), which allows to extract uniquely the dynamical system describing each cluster. The study of resonant clustering yielded a model of laminated turbulence [11] which gives a kinematic explanation of co-existence of STW and DWT in turbulent wave systems. These and other results, both for 3and 4-wave systems, are reviewed below, as well as their physical relevance and application areas. (a)E-mail: [email protected] Why are predictions of SWT theory often not corroborated? – The SWT theory assumes weak nonlinearity, randomness of phases and infinite-box limit, i.e. the resonance broadening Ω is greater than the spacing δω between adjacent wave modes