Abstract:

I discuss how nonlinear pdes are either isospectral or nonisospectral. The isospectral problems are integrable by the inverse scattering transform (here assumed to have spatially periodic or quasiperiodic boundary conditions). The nonisospectral problems are not integrable in any known sense, but occur in a vast number of important physics problems. The beauty of the isospectral problems is that they have very nice properties, including the fact that they are associated with a Riemann surface, and have a Riemann spectrum and particular phases. Riemann theta functions provide the actual solutions of these equations and form the basis of the "hyperfast" numerical methods. Theta functions are multidimensional Fourier series that provide for the possibility of describing periodic/quasiperiodic solutions of nonlinear pdes with coherent structures, i.e. with solitons, vortices, shocks, unstable wave packets (where rogue waves live), together with the usual Stokes waves and sine waves. The theoretical formulation reduces to linear Fourier analysis in the small amplitude limit. Codes of this type are "perfectly parallel" because the theta functions are linear and therefore huge speedups are possible on massively parallel systems. I discuss how the Riemann spectrum can be derived to include the coherent structures. This means that one can (1) develop time series analysis algorithms, (2) construct hyperfast models and (3) conduct nonlinear filtering simply as a consequence of the nice properties of theta functions. I discuss particular applications for laboratory experiments on random rogue wave trains and their nonlinear time series analysis, the nonlinear modeling of shallow water waves and the forensic studies of sunken ships.[1] In all of this work the Riemann theta function is the order of the day and provides the essential tool.

Nonisospectral problems are a particular challenge. Are the methods of algebraic geometry used above still useful for this most important class of nonlinear pdes? I give a scenario that suggests that this is so and discuss some of the details.[2] Development work is underway for (1) a coastal dynamics model over variable bathymetry using the Boussinesq equations, (2) a forth generation wind/wave spectral model with the full (hyperfast) Boltzmann integral providing nonlinear interactions and (3) the Euler equations for surface water waves. Computational fluid dynamical tools are also under development. Theta functions and the associated multidimensional Fourier series are the fundamental ingredients in the numerical codes.

[1] Alfred R. Osborne, The Sinking of the Prestige: Were Rogue Waves Involved?, Elsevier, 2012.

[2] Alfred R. Osborne, Rogue Waves and Holes in the Sea, Elsevier, 2012.