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Foundations of Integrability

Explain the meanings of integrability and the common mathematical language needed across the Library.

This volume will develop the subjects in the chapter map below. Use the outline to locate the methods and examples you want to study; the individual readings are still being developed.

For an available introduction, try What makes a system integrable?. The comparison reference states the classical definition and distinguishes it from quantum constructions and other uses of the term. These provide an entrance while this volume’s longer treatments are developed.

Readings and planned coverage

CHAPTER 01

Meanings of integrability

Planned coverage

  • Integrability versus exact solvability

CHAPTER 02

Conservation laws, symmetry & locality

Planned coverage

  • Local and quasilocal conserved charges

CHAPTER 03

Hamiltonian, Poisson & symplectic language

Planned coverage

  • Poisson brackets and Hamiltonian flows

CHAPTER 04

Linear spectral & scattering theory

Planned coverage

  • Spectral parameters and auxiliary linear problems

CHAPTER 05

Algebra, representations & operator language

Planned coverage

  • Tensor products and intertwiners

CHAPTER 06

Complex analysis, geometry & asymptotics

Planned coverage

  • Riemann surfaces for integrable systems

CHAPTER 07

Classical–quantum relations & limiting procedures

Planned coverage

  • Classical, continuum and thermodynamic limits

CHAPTER 08

Evidence, diagnostics & counterexamples

Planned coverage

  • What does a Lax pair establish?

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