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David Cai : Spatiotemporal Chaos, Weak Turbulence, Solitonic Turbulence & Invariant Measures - Statistical Characterization of Nonlinear Waves

We will present an overview of the program of statistical description of long time, large scale dynamics of nonlinear waves and highlight some results we obtained: from vanishing mutual information measures in the spatiotemporal chaos induced by hyperbolic structures of PDEs, to confirmation of weak-turbulence spectra, and role of coherent structures in controlling energy transfer in turbulent cycles described by multiple cascade spectra, to effective stochastic dynamics. We will address the issue of how to obtain invariant measures for these systems. Finally, we will report on our study of statistical properties of the focusing nonlinear Schrödinger equation, in the limit of a large number of solitons, corresponding to the semi-classical limit in a periodic domain. Our results demonstrate that the dynamics is described solitonic turbulence and there is a power law for the energy spectrum in the regime. We will discuss the connection between the wave turbulence and solitonic turbulence.

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