Chapter 16 - Nakamura-Boyd Approach

Abstract

Publisher Summary This chapter discusses how to determine the Riemann spectrum of the Korteweg-de Vries (KdV) equation with periodic boundary conditions using the method of Nakamura and Boyd. The Riemann spectrum is of course the natural, nonlinear generalization of the ordinary, linear Fourier spectrum for the KdV equation. The chapter describes certain aspects of the determination of the Riemann spectrum for numerical modeling and data analysis purposes. The method of approach is the so-called direct method or physical effectivization method. This method has been found to be quite useful in the time series analysis and numerical modeling of nonlinear wave trains. The direct method contrasts to the elegant algebrogeometric method found on algebraic geometry. Algebraic geometry provides two methods for determining the Riemann spectrum: (1) the loop integrals Schottky uniformization and (2) Schottky uniformization. The Nakamura–Boyd approach helps to compute the Riemann spectrum using a modification of theta functions called “theta functions with characteristics.” The chapter also describes how to compute the Riemann spectrum necessary to model waves using the Riemann theta functions. This step of the Nakamura–Boyd approach parallels the substitution of the linear Fourier transform into a linear Partial Differential Equation (PDE) to determine the linear-dispersion relation. The strategy for determining the solutions of nonlinear equations are also discussed.