Georgia Institute of Technology College of Sciences

Rogue waves are unusually large waves that appear from nowhere at the ocean. In the last 10 years or so, they have been the subject of numerous studies that propose homoclinic orbits of the NLS equation, the so-called breathers, to model such extreme events. Clearly, the NLS equation is an asymptotic approximation of the Euler equations in the spectral narrowband limit and it does not capture strong nonlinear features of the full Euler model. Motivated by the preceding studies, I will present recent results on deep-water modulated wavetrains and breathers of the Hamiltonian Zakharov equation, higher-order asymptotic model of the Euler equations for water waves. They provide new insights into the occurrence and existence of rogue waves and their breaking. Web info: http://arxiv.org/abs/1309.0668

The main aim of this talk is to design efficient and novel numerical algorithms for highly oscillatory dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become very inefficient when the longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. The framework of the heterogeneous multiscale method (HMM) will be considered as a general strategy both for the design and for the analysis of multiscale methods.(Keywords: Multiscale oscillatory dynamical systems, numerical averaging methods.)

I will first give a brief review on simple and robust central-upwind schemes for hyperbolic conservation laws. I will then discuss their application to the Saint-Venant system of shallow water equations. This can be done in a straightforward manner, but then the resulting scheme may suffer from the lack of balance between the fluxes and (possibly singular) geometric source term, which may lead to a so-called numerical storm, and from appearance of negative values of the water height, which may destroy the entire computed solution. To circumvent these difficulties, we have developed a special technique, which guarantees that the designed second-order central-upwind scheme is both well-balanced and positivity preserving. Finally, I will show how the scheme can be extended to the two-layer shallow water equations and to the Savage-Hutter type model of submarine landslides and generated tsunami waves, which, in addition to the geometric source term, contain nonconservative interlayer exchange terms. It is well-known that such terms, which arise in many different multiphase models, are extremely sensitive to a particular choice their numerical discretization. To circumvent this difficulty, we rewrite the studied systems in a different way so that the nonconservative terms are multiplied by a quantity, which is, in all practically meaningful cases, very small. We then apply the central-upwind scheme to the rewritten system and demonstrate robustness and superb performance of the proposed method on a number numerical examples.

I will present some results on the space-time stereo reconstruction of nonlinear sea waves off the Venice coast using a Variational Wave Acquisition Stereo System (VWASS). Energy wave spectra, wave dispersion and nonlinearities are then discussed. The delicate balance of dispersion and nonlinearities may yield the formation of solitons or traveling waves. These are introduced in the context of the Euler equations and the associatedthird order compact Zakharov equation. Traveling waves exist also in the axisymmetric Navier-Stokes equations. Indeed, it will be shown that the NS equations can be reduced to generalized Camassa-Holm equations that support smooth solitons and peakons.