Resonance Systems with Modulational Instability (RSMI)

This project deals with energy cascade generation in nonlinear system of surface water waves under narrow initial excitation. This problem is of the utmost importance both for laboratory experiments and for real physical phenomena, e.g. frequency downshifting, wave breaking, freak waves nonlinear excitation. Surface water waves are usually studied in the frame of kinetic wave turbulence theory, with description basing on statistical approach and resulting in power energy spectra. However, in a wave system with a narrow initial excitation statistical description does not describe the wave field evolution satisfactory, at least at the initial stage with just a few energetically active modes. Quite recently, a novel qualitative model of dynamical energy cascade in 3- and 4-wave systems with narrow initial excitation has been proposed, based on 1) resonance clusters of special structure, and 2) modulational instability as first step of a dynamical cascade. This cascade is described by a dynamical system (that is, no statistical assumptions are needed) and yields a discretized energy spectra of a certain structure. Contrary to the statistical description, the very appearance and various characteristics of dynamical energy cascade depend, first of all, on the initial and boundary conditions of the wave systems. Besides, such ground-breaking phenomena as wave propagation in inhomogeneous medium, dissipation rates and wave breaking can substantially modify cluster dynamics. The overall goal of this project lies in systematic theoretical, numerical and laboratory study of (1) resonance clustering originating from modulational instability, (2) non-conservative effects such as wave breaking and wind pumping, (3) current`s impact on nonlinear wave propagation. The expected outcome of the project is a description of 1. A basic model describing discretized energy spectrum in a 4-wave system of water waves; 2. Energy and momentum dissipation rates due to wave breaking; modified basic model (pumping and dissipation included); 3. Blocking phenomenon in the case of opposite current; effect of the initial steepness; modified basic model (wave-current interactions included); 4. Conditions of phase synchronization in a cascading cluster (a possible mechanism for freak wave formation), at least in the frame of the basic model.

The project leader has developed a novel model, called dynamical energy cascade, which describes an energy transfer in nonlinear systems not statistically but dynamically, as a resonance clusters formed by modulation instability. This model has been developed as a replacement for the conventional statistical theory, called kinetic wave turbulence theory (KWTT) which proved to be untenable by numerous experimental research. Dynamical energy cascades have been studied in various physical systems i.e. described by the nonlinear Schrödinger equation and its modifications, and also by the family of Korteweg-de Vries equations with different nonlinear terms. The following universal features of dynamical cascades have been established:-Time scale for the appearance of dynamical cascades corresponds observations and differs substantially from the time scale of KWTT. E.g. formation of a dynamical cascade among the surface water waves with the lengths of order 1 m takes about 2 minutes and can easily be observed experimentally, unlike kinetic energy cascade which in this case takes more than 2 years to occur.-The shape of energy spectrum depends on the parameters of excitation, such as frequency, amplitude, etc. The increment chain equation method (ICEM) has been specially developed for computing the shape of energy spectrum. Dissipation/forcing can easily be included into the general computation method.- The ICEM can be used both for 3- and 4-wave systems, i.e. the energy spectra observable experimentally in 3-wave systems are due to 4-wave interactions. This fact is confirmed e.g. in the experiments with capillary water waves. As to surface water waves, the model of dynamical cascade is the only know theoretical model which reproduces the celebrated Phillips spectrum and numerous results of experimental studies.- The first step of the dynamical cascade is described by the eminent exact solution of the nonlinear Schrödinger equation, met in many physical systems, namely, by the simplest form of the Akhmediev breather. The chain equation underlying the ICEM can be regarded as an exact resonance of the four Stokes modes, under some simplifications.- The model of the dynamical cascade allows to construct new connections among other known models in the theory of nonlinear waves, that is, connections that otherwise are neither known nor obvious. Accordingly, the model can be used as a common base for discussing results from different areas of nonlinear sciences.