The area of inverse scattering transform method or soliton theory has evolved over the past two decades in a vast variety of exciting new algebraic and analytic directions and has found numerous new applications. Methods and applications range from quantum group theory and exactly solvable statistical models to random matrices, random permutations, and number theory. The theory of isomonodromic deformations of systems of differential equations with rational coefficents, and mostnotably, the related apparatus of the Riemann-Hilbert problem, underlie the analytic side of this striking development. The contributions in this volume are based on lectures given by leading experts at the CRM workshop (Montreal, Canada). Included are both survey articles and more detailed expositionsrelating to the theory of isomonodromic deformations, the Riemann-Hilbert problem, and modern applications. The first part of the book represents the mathematical aspects of isomonodromic deformations; the second part deals mostly with the various appearances of isomonodromic deformations and Riemann-Hilbert methods in the theory of exactly solvable quantum field theory and statistical mechanical models, and related issues. The book elucidates for the first time in the current literature theimportant role that isomonodromic deformations play in the theory of integrable systems and their applications to physics.

Based on a series of graduate lectures, this book provides an introduction to algebraic geometric methods in the theory of complex linear differential equations. Starting from basic notions in complex algebraic geometry, it develops some of the classical problems of linear differential equations. It ends with applications to recent research questions related to mirror symmetry. The fundamental tool used is that of a vector bundle with connection. The book includes complete proofs, and applications to recent research questions. Aimed at graduate students and researchers, the book assumes some familiarity with basic complex algebraic geometry.

In 438 alphabetically-arranged essays, this work provides a useful overview of the core mathematical background for nonlinear science, as well as its applications to key problems in ecology and biological systems, chemical reaction-diffusion problems, geophysics, economics, electrical and mechanical oscillations in engineering systems, lasers and nonlinear optics, fluid mechanics and turbulence, and condensed matter physics, among others.

A thorough introduction to tau functions, from the basics through to the most recent results, with applications in mathematical physics.

The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also because it satisfies the aesthetic need in a beautiful formula which is so inherent to mathematics. The second edition is up-to-date and differs from the first one considerably. One third of the book (five chapters) is completely new and the rest is refreshed and edited. Contents: Integrable Systems Generated by Linear Differential n th Order Operators; Hamiltonian Structures; Hamiltonian Structure of the GD Hierarchies; Modified KdV and GD. The KupershmidtOCoWilson Theorem; The KP Hierarchy; Baker Function, a-Function; Additional Symmetries, String Equation; Grassmannian. Algebraic-Geometrical Krichever Solutions; Matrix First-Order Operator, AKNS-D Hierarchy; Generalization of the AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies; Isomonodromic Deformations and the Most General Matrix Hierarchy; Tau Functions of Matrix Hierarchies; KP, Modified KP, Constrained KP, Discrete KP, and q -KP; Another Chain of KP Hierarchies and Integrals Over Matrix Varieties; Transformational Properties of a Differential Operator under Diffeomorphisms and Classical W -Algebras; Further Restrictions of the KP, Stationary Equations; Stationary Equations of the Matrix Hierarchy; Field Lagrangian and Hamiltonian Formalism; Further Examples and Applications. Readership: Applied mathematicians and mathematical physicists."

In mathematical physics, one of the fascinating issues is the study of integrable systems. In particular, non-perturbative techniques that have been developed have triggered significant insight for real physics. There are basically two notions of integrability: classical integrability and quantum integrability. In this book, the focus is on the former, classical integrability. When the system has a finite number of degrees of freedom, it has been well captured by the Arnold–Liouville theorem. However, when the number of degrees of freedom is infinite, as in classical field theories, the integrable structure is enriched profoundly. In fact, the study of classically integrable field theories has a long history and various kinds of techniques, including the classical inverse scattering method, which have been developed so far. In previously published books, these techniques have been collected and well described and are easy to find in traditional, standard textbooks. One of the intriguing subjects in classically integrable systems is the investigation of deformations preserving integrability. Usually, it is not considered systematic to perform such a deformation, and one must study systems case by case and show the integrability of the deformed systems by constructing the associated Lax pair or action-angle variables. Recently, a new, systematic method to perform integrable deformations of 2D non-linear sigma models was developed. It was invented by C. Klimcik in 2002, and the integrability of the deformed sigma models was shown in 2008. The original work was done for 2D principal chiral models, but it has been generalized in various directions nowadays. In this book, the recent progress on this Yang–Baxter deformation is described in a pedagogical manner, including some simple examples. Applications of Yang–Baxter deformation to string theory are also described briefly.