Preface Foreword 1. Bilinearization of soliton equations 2. Determinants and pfaffians 3. Structure of soliton equations 4. Backlund transformations Afterword References Index.

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The direct method in soliton theory

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

Optical Waveguide Theory

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

Bäcklund transformations and their applications

Backlund and Darboux Transformations

Generalized Lax equations are considered in the spirit of Sato theory. Three decompositions of an underlying algebra of pseudo-differential operators lead, in turn, to three different classes of integrable nonlinear hierarchies. These are associated with Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili and Dym hierarchies in 2 + 1 dimensions. Miura- and auto-Backlund transformations are shown to originate naturally from gauge transformations of the Lax operators. General statements on reciprocal links between these hierarchies are established, which, in particular, give rise to novel reciprocal auto-Backlund transformations for the Dym hierarchy. These links are formulated as Darboux theorems for the associated Lax operators.

Colin Rogers

Papers

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

Bäcklund transformations and their applications

Backlund and Darboux Transformations

Gauge transformations and reciprocal links in 2 + 1 dimensions

Nonlinear Boundary Value Problems in Science and Engineering