Recommendations relating integrability and Lax potentials

Question

I was wondering if anybody can recommend notes/books that discusses integrability (i.e. models that possess as many conserved quantities as its number of degrees of freedom)? I'm particular interested to better understand the Lax potential and understand the maths behind the so-called graded \(\mathrm{sl}(2)\) loop algebra.

Comments

I'm not sure if this touches what you're looking for, but on the topic of integrability, I recall reading Gleb Arutyunov's pedagogical notes ("students seminar"), available online at http://www.staff.science.uu.nl/~aruty101/teaching.htm

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@Siva thanks for your answer. These notes certainly look interesting.

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Siva's link is now dead but the PDF in question can be found here http://www.astrology-tantra.com/BOOK/linked/1905.pdf or read here https://www.yumpu.com/en/document/view/26618414/student-seminar-classical-and-quantum-integrable-systems ;

Answer 1

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 e.g. these lecture notes, Section 3 of this article, and references therein; in the case of three independent variables see Section 3 of this same article,  subsubsection 10.3.3 of this book, and these lecture notes, and references therein; for the case of four independent variables see introductory part of this article, the same book as before, and references therein. The book in question in fact is a good introduction to integrable partial differential systems in general.

Answer 2

An excellent book on classical integrability, available online, is 

O Babelon, D Bernard, M Talon, Introduction to classical integrable systems, Cambridge University Press 2003.

Comments

Thank you; this seems very useful!!