Riemann Theta Functions: Applications to Hydrodynamical Flows and Ocean Waves
Speaker: Alfred R. Osborne, Nonlinear Waves Research Corporation
Date: Thursday, 24th September
Time: 5.30pm (Tea and coffee will be served from 5pm)
Location: A005, Health Science.
Abstract:
Nonlinear Waves Research Corporation
Riemann theta functions have been known since the 1850s, and most of the work on them over the past century and a half has been in the community of pure mathematics, most notably in the field of algebraic geometry. Slowly, however a number of fundamental results have occurred that are allowing theta functions to be used more and more for physical applications. My own work has been primarily in the area of ocean waves, both deterministic and stochastic, and both formulations are conveniently cast in terms of theta functions. Many applications have occurred in a number of fields, including ocean waves, nonlinear optics, hydrodynamics of fluids and plasmas, and string theory. I discuss the historical background of research on theta functions and come forward in time to modern studies of completely integrable Hamiltonian (soliton) systems. In this vein a fundamental result is that systems of this type contain coherent structure solutions in terms of Riemann theta functions. Perturbing systems of this type provides a rigorous basis for a kind of KAM Theory for nonlinear integrable and nearly integrable nonlinear partial differential equations, and I will give a particular example for the perturbed Korteweg-deVries equation.
One of the surprising results about the mathematical structure of theta functions is that they are useful for analyzing ocean wave data, and indeed many other data sets such as those that occur in nonlinear optics. A huge amount of work has been conducted over the past 35 years on numerical methods for dealing with theta functions and their support structure. One can determine when and what coherent structures occur in a particular data set, even when the waves are random in appearance. I give examples of both solitons and breather solutions using theta functions and discuss how these measured wave trains may be interpreted as “soliton and breather turbulence.” I show how one can develop the mathematical and numerical tools for using theta functions to study systems of this type, particularly to provide the capability of computing theta functions out to about genus 1000.
Seminar: Wave Group Seminar
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