# What is a good introduction to integrable models in physics?

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I would be interested in a good mathematician-friendly introduction to (exactly solvable)  integrable models in physics, either a book or expository article.

Related MathOverflow question: what-is-an-integrable-system.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Phira

edited May 4, 2014
Do you have anything specific in mind? I think the term integrability is sometimes used in slightly different contexts.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Pieter Naaijkens
The fact that "integrability" can mean so many things sometimes makes the quest to learn about it so challenging! I have found the introductory sections of Etingoff's paper www-math.mit.edu/~etingof/zlecnew.pdf to be a very good mathematical reference for a particular, physically interesting system (Calogero-Moser) which describes particles interacting in one-dimension.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Eric Zaslow

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I take "integrable models" to mean "exactly solvable models in statistical physics".

You can take a look at the classic book

Otherwise this new book is quit readable and covers more than just solvable models

Others can probably give you more mathematician-friendly references, but I think it would be good if you could be more specific about what you are looking for.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Heidar
answered Sep 22, 2011 by (855 points)
Yes, "exactly solvable" is what I mean. Thanks, I will update my question later.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Phira
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This is a late response. Here are some more

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Vijay Murthy
answered May 6, 2012 by (90 points)
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Another good recent book:

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user just-learning
answered Nov 16, 2013 by (95 points)

Also see his lecture notes on integrability.

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These lecture notes by Babelon could be quite helpful.

answered Aug 31, 2018 by (40 points)

There is a much more detailed ''Introduction to classical integrable systems'' by O. Babelon, D. Bernard, and M. Talon, very much recommended!

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The slides from the lectures on integrable systems by Alexei Bolsinov from this course could be quite helpful (to get to the slides click on the Files tab at the above link). Also see the lecture notes by Boris Dubrovin "Integrable Systems and Riemann Surfaces".

answered Sep 5, 2018 by (0 points)
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Specifically for integrable partial differential systems which are dispersionless a.k.a. hydrodynamic-type, i.e. can be written as quasilinear first-order homogeneous systems, in the case of two independent variables see these lecture notes, Section 3 of this article, and references therein; in the case of threeindependent variables see Section 3 of this same article,  subsubsection 10.3.3 of the Dunajski book mentioned by @just-learning, and these lecture notes, and references therein; for the case of four independent variables see introductory part of this article, the Dunajski book again, and references therein.

answered Sep 7, 2018 by (40 points)

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