Dynamic Energy Cascades formed by Nonlinear Waves

One of the most fundamental questions to be answered about the behavior of a nonlinear wave system is: what are general features of the energy transport within the system? How are they depending on the initial energy of the wave field, on time and on a huge amount of other intrinsic parameters? Etc. As the energy of a wave is proportional to the square of its amplitude, we can reformulate these questions in a very simple form, taking as an example the waves on the water surface, which saw everyone - no matter whether sitting in your own kitchen looking at the boiling water in a pot, or on the beach watching the gusts of wind blustering over ocean. The possible and obviously important questions are: can this wind generate a 10-meter high wave or higher? How the moving direction of a wave depends on the wind direction? What are the conditions that the wave reaches the shore, and what is its amplitude at this point? And many others. The principal investigator of this project developed a novel model of the dynamical energy cascade (D-cascade) allowing to answer many questions similar to the formulated above in the physical systems possessing modulation instability. The model allows to compute the shape of energy spectra for the case of narrow initial excitation of the system (or, coming back to the example above, to compute the amplitudes of all waves which grew under the action of the wind with the specified parameters). This model has been used for explaining both experimental observations and results of numerical simulations with various types of water waves (capillary, gravity-capillary and gravity surface waves) governed by the classical energy-conserving nonlinear Schrödinger equation (NLS). In the present project, our research aims are mainly focused on the study of some general properties of the D-cascades along the following lines: (1) Influence of dissipation on the D-cascade shape in the parametrical NLS appearing in nonlinear optical waves; (2) Analytical study of D-cascades in the systems governed by the higher order Korteweg-de Vries equations; (3) Connections of the D-cascade model to some well-known models in the theory of nonlinear waves. For computing the shape of the D-cascade we plan to use the increment chain equation method (developed by the principal investigator of this project) and the conventional mathematical analysis of the partial differential equations. The expected outcome of our project is twofold: In addition to the important theoretical developments, the foreseen results can be used directly for designing a new optical dynamic media with the desired properties.