Abstract: This chapter discusses the Riemann Theta functions. It focuses on the applications of theta functions as tools for solving integrable, nonlinear wave equations for the analysis of data and for hyperfast modeling. The Riemann theta function requires a symmetric Riemann matrix. The reasons are related to a fundamental problem in pure and applied mathematics, but it can be easily seen why symmetry of the Riemann matrix is natural. The sum over the symmetric part is finite. Statistical properties of Theta function parameters are presented. The chapter elaborates on the ordinary Fourier representation for the theta function. Gaussian series (also known as Poisson summation) are also useful for computing theta functions. The chapter discusses the one-dimensional case and addresses N degrees of freedom. The infinite-line limit occurs for only two Gaussians; the overlapping tails of the two Gaussians give the soliton after taking the second derivative of the log of the sum of the Gaussians. Solitons on the infinite line and on the periodic interval are also described.
Publication Year: 2010
Publication Date: 2010-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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