Discrete Resonances in Nonlinear Wave Systems (DRNWS)

This project deals with the problem of pattern formation in nonlinear dispersive wave systems governed in general by Euler equations. If nonlinearity is small (in a certain well-defined sense), wave system demonstrates stochastic behavior and is described by the wave kinetic equation on the corresponding time scale. Under certain restrictions, that is, when a wave system has zero or periodic boundary conditions, its behavior is not pure stochastic anymore and formation of some regular patterns (clusters of resonantly interacting waves) is possible. This fact is due to the discreteness of the wave-vector spectrum, and patterns are then small groups of waves exhibiting often also periodic energy exchange among the waves of a group. In many nonlinear wave systems statistical and discrete dynamics co-exist, while there also exist wave systems where only discrete dynamics is of importance. Special numerical algorithms have been developed recently to compute discrete resonances for a big class of dispersive wave systems. It is also known that resonances play major role in short-term forecast of wave field evolution in these systems (for instance, for ocean surface waves). This is our motivation for deep study of the resonances` structure. The overall goal of this project is to investigate the structure of discrete clusters formed by resonances, focusing on the case of 2-dimensional dispersive waves, some examples being gravity surface wave, ocean and atmospheric planetary waves, drift waves in plasma, etc. We plan to develop a classification of possible patterns formed from primary elements, analogous to diagrams technique in quantum mechanics. We are also interested in the properties of discrete quasi-resonances which satisfy resonance conditions with a given non-zero resonance width. More precisely, we are going to write out explicitly necessary conditions for discrete quasi-resonances to be able to start, as determined by the form of dispersion function. This problem is of major importance for understanding of interplay between discrete and statistical dynamics. The expected outcome of the project is: (1) general classification of discrete patterns appearing due to exact resonances; (2) development of a special algorithm for computing all different patterns and corresponding dynamical systems; (3) description of main characteristics of resonance cascade (spectrum anisotropy, pure scale cascades, mixed scale- angle cascades, etc.); (4) obtaining the constructive estimation, global and local, of resonance width for quasi-resonances to be able to start, for different types of waves; (5) creation of a program tool in MATHEMATICA for studying resonances in nonlinear wave systems. The results will also be interesting with respect to applications as a basic model for description of some known physical phenomena (for instance, intra-seasonal oscillations in the Earth atmosphere).

In the frame of this project E. Kartashova has developed a general theory of nonlinear resonances. This theory can be employed in various application areas and is systematically presented in her book "Nonlinear Resonance Analysis: Theory, Computation, Applications", published by the well-known publishing house Cambridge University Press. The phenomenon of resonance has been first described mathematically by the well-known French mathematician H. Poincaré in the 19th century. The problem of dealing with differential equations possessing resonances has H. Poincaré appraised as "the fundamental problem of dynamics". This problem originates from the area of differential equations but their form depends on the solution of certain Diophantine equations which have to be solved first, by methods of number theory. Kartashova`s theory allows to find resonances effectively; the method used does not depend on the application area from which a differential equation originates, e.g. physics, chemistry, medicine or economics. The two most important application results of this project are development of the model of intra-seasonal oscillations in the Earth`s atmosphere (essential for climate predictability) and discovery of discrete turbulent regimes in wave systems (paramount for understanding the results of various laboratory experiments in hydrodynamics). A further possible application of this theory lies in developing of a mathematical model for huge (or freak) waves` formation in the oceans. Another important aspect of this work is developing of an interactive on-line encyclopedia on nonlinear resonances, including in-built computer programs for studying of illustrative examples of resonances. On-line encyclopedia is an important addendum for the course on the theory of nonlinear resonances, hold by E. Kartashova at J. Kepler University, Linz, since 2005.