Nonlinear resonance analysis is a unique mathematical tool that can be used to study resonances in relation to, but independently of, any single area of application. This is the first book to present the theory of nonlinear resonances as a new scientific field, with its own theory, computational methods, applications, and open questions. The book includes several worked examples, mostly taken from fluid dynamics, to explain the concepts discussed. Each chapter demonstrates how nonlinear resonance analysis can be applied to real systems, including large-scale phenomena in the Earth's atmosphere and novel wave turbulent regimes, and explains a range of laboratory experiments. The book also contains a detailed description of the latest computer software in the field. It is suitable for graduate students and researchers in nonlinear science and wave turbulence, along with fluid mechanics and number theory. Color versions of a selection of the figures are available at www.cambridge.org/9780521763608.

We suggest a way of rationalizing an intra-seasonal oscillations (IOs) of the Earth atmospheric flow as four meteorological relevant triads of interacting planetary waves, isolated from the system of all the rest planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of IOs both in the North and South Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.Introduction. Concept of mesoscopic systems most often appears in condensed matter physics, e.g, in studying properties of superconductors on a scale comparable with that of the Cooper pairs[1], of miniaturized transistors on a computer chip, of disordered (glassy, granular) systems, when self-averaging is inefficient and fluctuations or the system prehistory become important. Similar situation occurs also in various natural phenomena -from wave turbulent systems in the Ocean [2] and Atmosphere, when wave lengthes are compatible with the Earth radius [3], to medicine [4], and even in sociology and economics, when finite size of a system (population, sociological group, market) becomes important [5]. Mesoscopic regimes are at the frontier between detailed, dynamical and statistical, self-averaging description of systems. Important observation for finite-size, weakly-decaying wave systems was made in [6]: discrete spacial-time resonances form small isolated clusters of interacting modes without energy exchange between the clusters. Clearly, there is exists relatively short, "threshold" wave, involving into the cluster with size large enough to "penetrate" into region of very short waves, where statistical description, that ignores the resonance discreetness, is valid.

General properties of nonlinear wave interactions in systems with periodic boundary conditions (socalled resonators) are studied. The wave field evolution in resonators displays striking, peculiar characteristics because only a small number of waves dominate the energy transfer processes. Indeed, the very fact of setting up boundary conditions is more important than the specific form of the nonlinear evolution equation. Results are presented for a number of wave fields whose evolution is governed by different nonlinear equations: planetary waves, gravity waves, capillary waves, and drift waves in plasmas.PACS numbers: 47. l0.+g, 47.35.+i The resonant interaction theory (RIT) of wave propagation in dispersive systems has being intensively developed in recent years since the important role of resonant interactions among gravity waves on the surface of deep water was pointed out [1]. RIT is based on the idea that the linearized equation gives a suitable first approximation for sufficiently small finite-amplitude disturbances. Used together with the multiscale method, this idea leads to the study of the necessary kinematic conditions for resonant interactions of waves which takes the formwhere k; is the wave vector of the ith wave, cu; ta(k;) is its frequency, s is the number of resonantly interacting waves, and i 1,2, . . . , s. There is a common belief that "in many practical applications these equations are not sufficiently general, since a system that admits one resonant set of waves often admits many resonant sets simultaneously" ([2], p. 61). The corresponding system of coupled nonlinear differential equations describing the time evolution of the resonant wave field becomes very complex. The use of some additional assumptions about the wave spectrum (e.g. , that the wave fields are spatially homogeneous and Gaussian or that wave vectors are nearly collinear and equal in magnitudes, etc. ) allows us to replace this system by the so-called kinetic equations which govern the transfer of energy through the spectrum ([3-6]). All the investigations in these references have one feature in commonthey have been carried out for infinite domains, e.g. , on an infinite plane or an infinite channel. The result of this is the study of the wavelengths small enough in comparison to the size of the experimental basin; in these cases "reasonable agreement is normally obtained with theoretical results for the infinite case" ([7], p. 249). The cases when the wavelengths are comparable to the characteristic sizes of the experimental basins remain, as a rule, unexplained ([2) and bibliography therein) and are named effects of finite length" [7].The detailed study of these effects of finite length is our main goal. Namely, we study here the general properties of resonant wave interactions in systems with periodic or zero boundary conditions, i.e. , in resonators. This leads to the necessity of ftnding the integer solutions of the system (I). Thus, we consider below only wave vectors k with integer components.The interactions are l...

In this Letter we present 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 handy graphical presentation 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.1. Introduction. In 1960-1980th a great volume of research has been performed in the area of wave interactions [1], comprehensive account of theory and experiment can be found in [2]. The notion of 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 the immediate 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 complete resonance set, classical resonance curves [9] are not anymore a suitable representation, of course. 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 kinematic explanation of co-existence of STW and DWT in wave turbulent systems. These and other results, both for 3-and 4-wave systems, are reviewed below, as well as their physical relevance and application areas. 2. 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 δ ω between adjacent wave modes