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.
Topics in Soliton Theory, 1st Edition
KdV and KP; Analytic Methods.
Inverse Scattering. KdV on the Line. Problems in Mechanics and Hill's Equation. On the Geometry of KdV. Finite Zone Potentials. Hamiltonian Theory for KdV. Determinant Methods for KdV and KP; Tau Functions. Systems and Algebraic Methods.
Orbits of the Vacuum. AKNS Systems. Some Lie Theoretic Methods. The Hirota Bilinear Identity. Algebraic Curves and KP. Introductory Sato Theory. Physics.
Holonomic Quantum Fields. Ising Model and Bose Gas. Some remarks on 2-D Quantum Gravity and KdV. Conformal Field Theory. More on Conformal Field Theory and Tau Functions. More on Kricever Data, Grassmannians, Curves etc. Remarks on Strings. More on Strings, Riemann Surfaces, and Tau Functions. Remarks on Tau Functions, Cauchy-Riemann Operators, and Determinant Bundles. Quantum Inverse Scattering. Appendices: Differential Geometry and Elementary Hamiltonian Theory. Riemann Surfaces and Algebraic Curves. References. Index.