Waves, Solitons and Turbulence in Optical Systems - Abstract
Tobisch (Kartashova), Elena
defppartial defpxpartial_x defpypartial_y defaalpha defbbeta defggamma defddelta deføomega defwtwidetilde defmsmedskip defbebeginequation defbabeginarray defeeendequation defeaendarray defbea begineqnarray defeea endeqnarray defbeanbegineqnarray* defeeanendeqnarray* Wave turbulence theory (WTT) describes a temporal behavior of a system of weakly nonlinear dispersive waves of the form $A(T)expi[mathbfxcdot mathbfk -omegacdot t]$ with dispersion relation $omega=omega(mathbfk)$ and slowly changing amplitude $A(T)$ ($T$ is introduced emphvia a small parameter). This allows to reduce the study of these wave systems to the study exact resonances satisfying resonance conditions $ omega_1+omega_2=omega_3, quad mathbfk_1+mathbfk_2=mathbfk_3 $ (in the simplest case of 3-wave interactions). The wave system's time evolution can be described then by a set of independent dynamical systems (discrete WTT) built from the systems of the form be label1 dotA_1= V^3_12 A_2^*A_3,quad dotA_2= V^3_12 A_1^* A_3, quad dotA_3= - V^3_12 A_1 A_2, % ee %% or by the wave kinetic equation (kinetic WTT) $ fracbf d bf d tlangleA^2_3rangle=$ bea label3kin int V^3_12 ^2 delta(o_3-o_1-o_2)delta(mathbfk_3-mathbfk_1-mathbfk_2) cdot (A_1A_2-A_1^*A_3-A_2^*A_3) bf dmathbfk_1 bf dmathbfk_2. eea Detuned resonances satisfying $ omega_1+omega_2-omega_3 = Omega >0 $ are not regarded in this framework due to the following assumptions: (a) the input of detuned resonances in general energy balance is negligible, and (b) they take place at some later time scale. Quite recently, both of these allegations (a),(b) have been challenged and a so-called generalized wave kinetic equation was deduced which is based on detuned resonances and produces, emphin some cases, the time evolution similar to the Eq. (ref3kin) and at the "faster" time scale. However, these numerical results are not conclusive and for some initial parameters they demonstrate evolution, qualitatively different from that predicted by the Eq. (ref3kin). In my talk I plan to discuss these issues and demonstrate that their cause lies in the fact that the standard definition of the frequency detuning is not unique, and in fact describes two physically different systems; this follows directly from the properties of the Sys. (ref1). I will also give examples of how to determine the frequency detuning uniquely, for a given dispersion relation $omega=omega(mathbfk)$.