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I am currently studying non-linearity properties of hydrodynamic waves from a purely mathematical stance. At the moment, I am concerned with radiation, modulational instability described by korteweg-De Vries equation (fifth order) and Spatial Instabilities and chaos in a high order hamiltonian algebra. These characteristics of radiating wave tails that propagate outwards in what seems to be a chaotic manner are my concern. I am elaborating my analysis to focus primarily on the characteristics of Spatially quasi-periodic capillary-gravity waves using the Hamiltonian structures which describes the degeneracy of the water-wave problem.

With this in mind, my observations ideas consist of a 'turbulent flow' of incompressible fluids travelling through a differentiable manifold and experiencing a Reynold Stress on it's surface (vorticity is key),

How can I adapt a system of 'eddies' in flux with stochastic calculus (brownian motion). e.g: If I were to study a batch of wave propagation in a free surface, Is it possible for me to derive a system that can equally predict and forecast the random and chaotic behavior of it's sub-atomic particles? Dissipation at the Quantum Level comes to mind.

Thank you!!

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# Non-Linear Water Waves

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