Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust analysis is proposed, and new criteria for when calculations should be terminated are introduced and compared with classical resolution studies and spectral convergence tests. Most of the analysis is on a 512 × 128 × 2048 mesh, with new analysis on a just completed 1024 × 256 × 2048 used to confirm trends. One might hypothesize that there is a finite-time singularity with enstrophy growth like Ω ∼ (Tc − t) −γ Ω and vorticity growth like ω ∞ ∼ (Tc − t) −γ . The new analysis would then support γΩ ≈ 1/2 and γ > 1. These represent modifications of the conclusions of Kerr (1993). Issues that might arise at higher resolution are discussed.

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to 4096 3 . The results are analyzed in terms of the classical analyticity-strip method and Beale, Kato, and Majda (BKM) theorem. A well-resolved acceleration of the time decay of the width of the analyticity strip δ(t) is observed at the highest resolution for 3.7 < t < 3.85 while preliminary three-dimensional visualizations show the collision of vortex sheets. The BKM criterion on the power-law growth of the supremum of the vorticity, applied on the same time interval, is not inconsistent with the occurrence of a singularity around t 4. These findings lead us to investigate how fast the analyticity-strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if δ(t) vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for δ(t) then its exponent must be greater than some critical value, thus providing a new test that is applied to our 4096 3 Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time interval on which a well-resolved power-law behavior of δ(t) takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay.

We apply Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider α and β Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We demonstrate that resonances are responsible for the irreversible transfer of energy among the Fourier modes. We predict that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity. Our methodology is not limited to only these systems and can be applied to the case of a finite number of modes, such as in the original FPUT experiment, or to the thermodynamic limit, i.e. when the number of modes approach infinity. In the latter limit, we perform state of the art numerical simulations and show that the results are consistent with theoretical predictions. We suggest that the route to thermalization, based only on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.

We consider the set of Diophantine equations that arise in the context of the partial differential equation called "barotropic vorticity equation" on periodic domains, when nonlinear wave interactions are studied to leading order in the amplitudes. The solutions to this set of Diophantine equations are of interest in atmosphere (Rossby waves) and Tokamak plasmas (drift waves), because they provide the values of the spectral wavevectors that interact resonantly via three-wave interactions. These wavenumbers come in "triads", i.e., groups of three wavevectors.We provide the full solution to the Diophantine equations in the physically sensible limit when the Rossby deformation radius is infinite. The method is completely new, and relies on mapping the unknown variables via rational transformations, first to rational points on elliptic curves and surfaces, and from there to rational points on quadratic forms of "Minkowski" type (such as the familiar space-time in special relativity). Classical methods invented centuries ago by Fermat, Euler, Lagrange, Minkowski, are used to classify all solutions to our original Diophantine equations, thus providing a computational method to generate numerically all the resonant triads in the system. Computationally speaking, our method has a clear advantage over brute-force numerical search: on a 10000 2 grid, the brute-force search would take 15 years using optimised C ++ codes on a cluster, whereas our method takes about 40 minutes using a laptop.Moreover, the method is extended to generate so-called quasi-resonant triads, which are defined by relaxing the resonant condition on the frequencies, allowing for a small mismatch. Quasi-resonant triads' distribution in wavevector space is robust with respect to physical perturbations, unlike resonant triads' distribution. Therefore, the extended method is really valuable in practical terms. We show that the set of quasi-resonant triads form an intricate network of connected triads, forming clusters whose structure depends on the value of the allowed mismatch. It is believed that understanding this network is absolutely relevant to understanding turbulence. We provide some quantitative comparison between the clusters' structure and the onset of fully nonlinear turbulent regime in the barotropic vorticity equation, and we provide perspectives for new research.

In this Letter we show how the nonlinear evolution of a resonant triad depends on the special combination of the modes' phases chosen according to the resonance conditions. This phase combination is called dynamical phase. Its evolution is studied for two integrable cases: a triad and a cluster formed by two connected triads, using a numerical method which is fully validated by monitoring the conserved quantities known analytically. We show that dynamical phases, usually regarded as equal to zero or constants, play a substantial role in the dynamics of the clusters. Indeed, some effects are (i) to diminish the period of energy exchange τ within a cluster by 20% and more; (ii) to diminish, at time scale τ , the variability of wave energies by 25% and more; (iii) to generate a new time scale, T >> τ , in which we observe considerable energy exchange within a cluster, as well as a periodic behaviour (with period T ) in the variability of modes' energies. These findings can be applied, for example, to the control of energy input, exchange and output in Tokamaks; for explanation of some experimental results; to guide and improve the performance of experiments; to interpret the results of numerical simulations, etc. PACS numbers: 47.27.Ak, 47.27.ed, 52.25.Fi 1. Introduction. Nonlinear resonances are ubiquitous in physics. Euler equations, regarded with various boundary conditions and specific values of some parameters, describe an enormous number of nonlinear dispersive wave systems (capillary waves, surface water waves, atmospheric planetary waves, drift waves in plasma, etc.) all possessing nonlinear resonances [1]. Nonlinear resonances appear in mechanics [2], astronomy [3], medicine [4], etc., etc. In this Letter we will regard the simplest nonlinear resonant systems corresponding to the 3-wave resonance conditions. Examples of these nonlinear resonant systems are, in order of simplicity, triads (which are integrable), and small groups of connected triads which are known to be important for various physical applications (large-scale motions in the Earth's atmosphere [5], laboratory experiments with gravity-capillary waves [6], etc.). Dynamical system for a triad will be regarded in the standard Manley-Rowe form:

It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We show that 1) most frequently met clusters are described by integrable dynamical systems, and 2) construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for system with cubic Hamiltonian. PACS numbers: 47.10.Df, 47.10.Fg, 02.70.Dh 1. Introduction. A notion of resonance runs through all our life. Without resonance we wouldn't have radio, television, music, etc. The general properties of linear resonances are quite well-known; their nonlinear counterpart is substantially less studied though interest in understanding nonlinear resonances is enormous. Famous experiments of Tesla show how disastrous resonances can be: he studied experimentally vibrations of an iron column which ran downward into the foundation of the building, and caused sort of a small earthquake in Manhattan, with smashed windows and swayed buildings [1]. Another example is Tacoma Narrows Bridge which tore itself apart and collapsed (in 1940) under a wind of only 42 mph, though designed for winds of 120 mph. Nonlinear resonances are ubiquitous in physics. Euler equations, regarded with various boundary conditions and specific values of some parameters, describe an enormous number of nonlinear dispersive wave systems (capillary waves, surface water waves, atmospheric planetary waves, drift waves in plasma, etc.) all possessing nonlinear resonances [2]. Nonlinear resonances appear in a great amount of typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate [3].The so-called "nonlinear resonance jump", important for the analysis of a turbine governor positioning system of hydroelectric power plants, can cause severe damage to the mechanical, hydraulic and electrical systems [4]. It was recently established that nonlinear resonance is the dominant mechanism behind outer ionization and energy absorption in near infrared laser-driven rare-gas or metal clusters [5]. The characteristic resonant frequencies observed in accretion disks allow astronomers to determine whether the object is a black hole, a neutron star, or a quark star [6]. Thermally induced variations of the helium dielectric permittivity in superconductors are due to microwave nonlinear resonances [7]. Temporal processing in the central auditory nervous system analyzes sounds using networks of nonlinear neural resonators [8]. The non-linear resonant response of biological tissue to the action of an electromagnetic field is used to investigate cases of suspected disease or cancer [9].

A robust energy transfer mechanism is found in nonlinear wave systems, which favors transfers toward modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We emphasize the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads' precession frequencies resonate with the system's nonlinear frequencies, leading to a collective state of synchronized triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations confirm analytical predictions.

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.