MATHEMATICS · PHYSICS · CONNECTIONS
A connected guide to
integrable systems.
Explore the mathematics and physics of integrable systems: a subject connecting mechanics, nonlinear waves, quantum many-body theory, geometry, and probability. Find a route through the ideas, a model to study, or a precise reference.
New to the subject? Start with What makes a system integrable?, then choose a concrete problem below.
A COMPLETE LEARNING SEQUENCE
Three particles. A whole theory.
Start with Hamilton’s equations. Build a Lax pair, find conserved quantities, prove integrability, and test a trajectory against an exact solution.
A QUANTUM LEARNING SEQUENCE
Two spin flips. An interacting wave.
Construct a spin-chain Hamiltonian. Follow a magnon, derive its scattering with a second, and test Bethe equations against a finite quantum system.
A NONLINEAR WAVE SEQUENCE
One pulse. A linear spectral problem.
Derive a solitary wave, find its conserved integrals, and reconstruct it from one bound state. Test its motion with a computation that separates time, space and boundary errors.
A PROBABILITY LEARNING SEQUENCE
Random hops. An exact current.
Build a Markov generator for particles on a ring. Derive its stationary current, reconstruct a Bethe decay mode, and compare trajectories with finite-state evolution.
SIX WAYS INTO THE SUBJECT
Where would you like to begin?
Build understanding
Classical, quantum, wave and probability routes, targeted preparation, and worked practice.
02 / LibraryExplore the theory
Sixteen connected volumes, from Hamiltonian mechanics and solitons to probability and quantum fields.
03 / ModelsStart with a system
Equations, regimes, and the precise sense in which a model is integrable or exactly solvable.
04 / ResearchFollow a question
A reproducible singular-Bethe-state case study, with a map of further research questions to develop.
05 / ReferenceFind a precise answer
Definitions, conventions, formulas, and results with their assumptions and sources.
06 / AboutUnderstand the project
The purpose, scope, and editorial principles of a resource being built for students and researchers.
CHOOSE A ROUTE, NOT A LADDER
Different backgrounds.
Connected ideas.
There is no need to read the entire library in order. Begin with a familiar problem and follow the mathematics it needs.
- From mechanics to classical integrability
Hamiltonian systems, actions and angles, Lax pairs, solitons.
- From quantum mechanics to the Bethe ansatz
Spin chains, scattering, commuting transfer matrices.
- From your field into the subject
Begin with finite-ring TASEP. Geometry, field theory and further specialist routes are planned.