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.
Learning sequences: Open Toda · The XXX spin chain · KdV solitons · Finite-ring TASEP
Chapter map
- Meanings of integrability
- Conservation laws, symmetry & locality
- Hamiltonian, Poisson & symplectic language
- Linear spectral & scattering theory
- Algebra, representations & operator language
- Complex analysis, geometry & asymptotics
- Classical–quantum relations & limiting procedures
- Evidence, diagnostics & counterexamples
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?