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Integrable Systems Reference

Find precise definitions, conventions, formulas, and results. The reference is organized for focused lookup, with assumptions and regimes kept beside the statements that depend on them.

A concise formula is useful only with its definitions and conditions. Begin with Definitions of integrability compared to distinguish classical hypotheses, quantum constructions, exact stochastic observables and numerical evidence. The other available references state the Liouville–Arnold hypotheses for invariant tori, explain Lax equations, distinguish solitary waves from solitons, translate spin-chain, R-matrix and KdV conventions, fix Markov generators and stochastic current definitions, and collect checked source editions with BibTeX downloads. Each links back to the relevant explanations and model definitions.

The chapter map also identifies future lookup material. Plain-text titles are planned entries, not available formulas or results.

Available readings

Readings and planned coverage

CHAPTER 01

Terms & Notation

Planned coverage

  • Glossary A–Z
  • Symbols by context
  • Abbreviations, aliases & spellings
  • Subject & object index

CHAPTER 02

Conventions & Dictionaries

Planned coverage

  • Hamiltonian & Poisson conventions
  • Fourier, convolution & distribution conventions
  • Units, rescalings & dimensionless parameters
  • Lax, monodromy & zero-curvature conventions
  • Spectral parameters, rapidities & branches
  • Bethe phases & state normalization
  • Root densities, fillings & dressing conventions
  • Periods, cycles, elliptic & theta conventions
  • Monodromy, Stokes & tau-function conventions

CHAPTER 03

Equations & Identities

Planned coverage

  • Equation & identity index
  • Hamiltonian & Poisson identities
  • Scattering & reconstruction relations
  • Yang–Baxter, RTT & reflection relations
  • Bethe, Baxter & fusion relations
  • Hirota bilinear & tau-function identities
  • Determinant, Pfaffian & Fredholm identities
  • S-matrix & form-factor relations
  • Thermodynamic & dressing equations
  • Hydrodynamic & continuity equations
  • Spectral curves & period relations
  • Monodromy & isomonodromic equations
  • Elliptic, theta & special-function identities

CHAPTER 04

Theorems & Criteria

Planned coverage

  • Theorem & criterion index
  • Hamilton–Jacobi & Stäckel criteria
  • Lax & classical r-matrix criteria
  • Commuting transfer-matrix criteria
  • Bi-Hamiltonian & recursion criteria
  • Conserved charges & factorized-scattering criteria
  • Multidimensional consistency criteria
  • KAM theorem: hypotheses & limits
  • Differential-Galois obstructions
  • Integrability diagnostics & their limits

CHAPTER 05

Regimes & Observables

Planned coverage

  • Boundary conditions, sectors & domains
  • Limits, scaling & order of limits
  • Approximations & breakdown conditions
  • Charges, densities & quasiparticles
  • Spectra, gaps & finite-size quantities
  • Correlations, response & spectral functions
  • Transport coefficients & counting statistics
  • Entanglement & information quantities
  • Soliton, scattering & wave observables
  • Probability laws & stochastic observables
  • Geometric, monodromy & period data

CHAPTER 06

Methods & Correspondences

Planned coverage

  • Method finder
  • Spectral & scattering methods compared
  • Bethe & functional-equation methods compared
  • Transformations & separation methods compared
  • Asymptotic & numerical methods compared
  • Classical–quantum & lattice–continuum dictionaries
  • Statistical–quantum & stochastic dictionaries
  • ODE/IM & isomonodromic dictionaries
  • Gauge-theory & geometric dictionaries

CHAPTER 07

Software, Notebooks & Data

Planned coverage

  • Software catalog
  • Notebooks & executable examples
  • Reference datasets & numerical tables
  • Benchmark specifications & test cases
  • Reproducible environments & run manifests
  • Numerical precision & error reporting
  • Artifact versions, provenance & citation

CHAPTER 08

Bibliography & Sources

Planned coverage

  • Books & monographs
  • Reviews & lecture notes
  • Original papers & result locators
  • Theses, courses & recorded lectures
  • Source concordances & edition differences
  • Errata, translations & version histories

CHAPTER 09

History & Common Confusions

Planned coverage

  • Historical timeline
  • Terminology & attribution history
  • Integrable, exactly solvable & free
  • Conservation, completeness & locality
  • Spectral data versus physical observables
  • Limits, diagnostics & equivalence pitfalls

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