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Library of Integrable Systems

Explore integrable systems in depth through sixteen subject volumes. Each volume provides a chapter map; the collection connects classical mechanics, nonlinear waves, quantum systems, geometry, probability, and applications.

The volumes are entrances, not a compulsory sequence. Current derivations establish finite open Toda integrability, construct XXX eigenstates and a complete four-site sector, prove the commuting-transfer-matrix mechanism, reconstruct one- and two-soliton KdV fields, and derive a two-particle Bethe mode for periodic TASEP. Find these readings directly below, before exploring the wider sixteen-volume map.

Choose a question or system first, then use the relevant mathematical tools. Learn provides guided calculations and practice; Models fixes the equations and regimes; Reference supplies convention translations and checked source editions. Unlinked chapter titles describe material still to be written.

Available readings

Explore the volumes

01 8 chapters

Foundations of Integrability

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

02 7 chapters

Classical Hamiltonian Systems

Develop finite-dimensional integrable dynamics, its constructions, global geometry and obstructions.

03 8 chapters

Solitons & Nonlinear Waves

Explain integrable nonlinear evolution equations, solution methods, initial data and wave phenomena.

04 8 chapters

Integrable Hierarchies & Geometry

Develop the geometric organization of commuting flows, spectral data and algebraically integrable systems.

05 7 chapters

Discrete Integrable Systems

Treat integrable maps, lattice equations, recurrences and discrete geometric structures as subjects in their own right.

06 9 chapters

Yang–Baxter Structures & Bethe Ansatz

Provide the reusable algebraic and spectral machinery used across quantum chains, statistical models, stochastic models and field theories.

07 7 chapters

Quantum Many-Body Systems

Develop the physical organization, construction and excitation content of integrable quantum matter.

08 7 chapters

Exactly Solvable Statistical Mechanics

Develop equilibrium lattice models, their solution structures, boundary dependence and critical limits.

09 8 chapters

Integrable Field Theory

Organize classical and quantum field theories around conserved currents, scattering, local operators and quantum consistency.

10 8 chapters

Thermodynamics & Correlation Functions

Turn spectral data into equilibrium states, finite-volume spectra, matrix elements and physical response.

11 8 chapters

Nonequilibrium Dynamics & Hydrodynamics

Explain relaxation, transport, evolving quasiparticle fluids and the limits of exact integrable dynamics.

12 8 chapters

Integrable Probability & Random Matrices

Develop exact distributions, stochastic evolution, random spectral objects and their scaling limits.

13 7 chapters

Isomonodromy & Special Functions

Develop monodromy-preserving problems, nonlinear special functions and exact differential-equation spectral methods.

14 7 chapters

Gauge Theory, Strings & Gravity

Explain integrable structures in supersymmetric gauge theories, strings, holography and special gravitational reductions.

15 7 chapters

Computational Methods

Teach reproducible calculations that turn formal exact descriptions into usable numerical or symbolic results.

16 6 chapters

Experiments & Applications

Connect physical realizations to effective integrable models, measurable observables and controlled limitations.

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