When soliton theory, based on water waves, plasmas, fiber optics etc., was developing in the 1960-1970 era it seemed that perhaps KdV (and a few other equations) were really rather special in the set of all interesting partial differential equations. As it turns out, although integrable systems are still special, the mathematical interaction of integrable systems theory with virtually all branches of mathematics (and with many currently developing areas of theoretical physics) illustrates the importance of this area. This book concentrates on developing the theme of the tau function. KdV and KP equations are treated extensively, with material on NLS and AKNS systems, and in following the tau function theme one is led to conformal field theory, strings, and other topics in physics. The extensive list of references contains about 1000 entries.
A main theme of the book outlines the role of the quantum potential in quantum mechanics and general relativity and one of its origins via fluctuations formulated in terms of Fisher information. Another theme is the description of various approaches to Bohmian mechanics and their role in quantum mechanics and general relativity. Along the way various approaches to, for instance, the Dirac equation, the Einstein equations, the Klein-Gordon equation, the Maxwell equations and the Schrödinger equations are described. Statistics and geometry are intertwined in various ways and, among other matters, the aether, cosmology, entropy, fractals, quantum Kaehler geometry, the vacuum and the zero point field are discussed. There is also some speculative material and some original work along with material extracted from over 1000 references and the work is current up to April 2005.
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