A. Integrable Systems

(1) Survey Articles

1. J. Moser, Various aspects of integrable Hamiltonian systems. Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978), pp. 233-289, Progr. Math. 8, Birkhauser, Boston, Mass.,1980. [This is a survey on various aspects underlying the integrability of a Hamiltonian system such as group representation, symplectic reduction and isospectral deformation. Several finite dimensional systems are discussed illustrating the various aspects. The survey ends with a section on the inverse spectral theory of the Hill's equation.]

2. H. McKean, Integrable systems and algebraic curves. Global analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Calgary, Calgary, Alberta, 1978), pp.83-200, Lecture Notes in Math., 755, Springer, 1979. [This is a detailed survey on the link between some special many-body problems and shallow water waves to algebraic geometry. The paper begins with the Fermi-Pasta-Ulam lattice and explains how it has led Kruskal and Zabusky to their numerical experiment. A good part of the paper is devoted to the KdV equation, in particular with regard to its solution in the periodic case and its connection to the Calogero-Moser systems. Other topics surveyed include: (a) a parallel theory for the periodic Toda lattice, (b) Boussinesq equation on a circle, (c) the Neumann system, (d) group-theoretic ideas.

3. P. Deift, Integrable Hamiltonian systems. Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994), 103-138, Lectures in Appl. Math. 31, Amer. Math. Soc., 1996. [This is a series of 4 lectures which aims to give a rapid introduction to the subject. The content includes a survey of the following topics: (a) notion of complete integrability and the Liouville-Arnold theorem, (b) the Toda lattice and its generalization, (c) classical r-matrices, (d) inverse scattering of the defocusing nonlinear Schrodinger (NLS) equation as a Riemann-Hilbert problem, (e) long time asymptotics of the defocusing NLS equation via the nonlinear steepest descent method.

4. A. Reyman and M. Semenov-Tian-Shansky, Group-theoretic methods in the theory of finite-dimensional integrable systems. Dynamical systems VIIm Encyclopaedia of Mathematical Sciences, (V.I. Arnold and S.P. Novikov, eds.), vol. 16, Springer-Verlag, 1994, pp.116- 225. [This is a survey of a general group-theoretic scheme (a.k.a. classical r-matrix theory) to construct integrable Hamiltonian systems and their solutions in a systematic way. Many finite dimensional examples are given to illustrate the general theory. At the end of each section, careful historical notes are given to guide the reader to the original literature.]

5. New developments in the theory and application of solitons (a discussion organized and edited by Sir Michael Atiyah, F.R.S., J.D. Gibbon, and G. Wilson), Phil. Trans. R. Soc. Lond. A, volume 315, pp. 333-469 (19850. [This includes contributions from J.D. Gibbon, J.B. Keller, P. van Moerbeke, I.B. Frenkel, G. Wilson, N. M. Ercolani and H. Flaschka, A.C. Scott, L.F. Mollenauer, R.S. Ward, Sir Michael Atiyah, F.R.S. and N.J. Hitchin, as well as a transcript of an interactive discussion moderated by J.T. Stuart, F.R.S. and M. Tabor]

6. Important developments in soliton theory, edited by A. Fokas and V. Zakharov, Springer, 1993. [This book consists of a collection of survey papers which covers the following aspects of the subject: (a) method of solution, (b) asymptotic methods, (c) algebraic aspects, (d) quantum and statistical mechanical methods, (e) near integrable methods and computational aspects]

7. A. Its, The Riemann-Hilbert problem and integrable systems, Notices of AMS, 50(11), 1389--1400, 2003. [This article presents some new developments in the Riemann-Hilbert formalism related to integrable systems using Painleve equations as a main example.]

(2) Texts

1. P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge University Press, Cambridge, 1989. [This is a short introductory text which covers the essence of soliton theory, including: the discovery of solitary waves, the inverse scattering method for the KdV equation, the AKNS-ZS scheme, Hirota's method, Painleve property, and numerical methods.]

2. A.C. Newell, Solitons in Mathematics and Physics, SIAM, 1985. [This is an introductory text on soliton theory that begins with a historical chapter followed by a chapter concerning the derivation of various soliton equations from physical models and a chapter about the inverse-scattering transform and its use (including some perturbation theory). This book then departs from this standard material, culminating with two chapters addressing the importance of symmetries in soliton theory, finally linking together Hirota's theory of tau-functions with the Wahlquist-Estabrook method and the theory of loop groups and vertex algebras.]

3. L. Faddeev and L. Takhtajan, Hamiltonian methods in the theory of solitons, Springer-Verlag, Berlin, 1987.[This book presents the Hamiltonian approach to soliton theory. The main point of view is that the soliton equations are infinite dimensional completely integrable Hamiltonian systems. The first part of the book covers the nonlinear Schrodinger equation in detail, including the formulation of the inverse problem as a Riemann-Hilbert problem and the construction of action-angle variables. In the second part, the authors apply the same ideas to other equations including : the Sine-Gordon equation, the Toda lattice, the Heisenberg ferromagnet and the Landau-Lifschitz equation. The book ends with a sketchy chapter on the Lie-theoretic approach to integrable models.]

(3) Papers

1. P. Lax, Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467-490 [This is a very significant paper in which the author introduced the well-known "Lax pair" for the KdV equation. Nowadays, the method of Lax pairs underlies much of the modern theory of integrable systems]

2. V. Zakharov, L. Faddeev, The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funct. Anal. Appl. 5, 280-287 (1971), translated from Funk. Anal. Priloz 5 (4), 16-27 (1971) [In this important paper, the authors showed that the variables which linearize the KdV equation can be interpreted as action-angle variables. Thus this establishes the KdV equation as an infinite dimensional integrable Hamiltonian system]

3. V. Zakharov, A. Shabat, Exact theory of two-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 62-69; translated from \v Z.Eksper. Teoret. Fiz. 61 (1971), no. 1, 118-134(Russian)81.35 [The authors showed that the nonlinear Schrodinger equation also has a Lax pair formulation. Prior to this work, it was not clear if there exist other equations besides KdV for which the Lax pair formulation is possible]

4. C. Gardner, J. Greene, M. Kruskal, R. Miura, Korteweg-de Vries equation and generalization VI. Methods for exact solution. Comm. Pure Appl. Math. 27 (1974), 97-133 [In a short note which appeared in 1967, the authors announced a nonlinear change of variables under which the dynamics of the KdV equation becomes linear. In this seminal contribution, the complete detail of the inverse scattering method was worked out in the context of the KdV equation. Subsequently, the method was found to be applicable to other important integrable PDEs and lattice models]

5. H. Flaschka, The Toda lattice I. Existence of integrals. Phys. Rev. B (3) 9 (1974), 1924-1925. [In this paper, the author made the important change of variable which allowed him to rewrite the equations of motion of the Toda lattice as an isospectral deformation of a Jacobi matrix. This has led to many subsequent developments in the area]

6. H. Flaschka, The Toda lattice II. Inverse scattering solution. Progr. Theoret. Phys. 51 (1974), 703-716. [In this work, an inverse scattering method is developed for the solution of the doubly-infinite Toda lattice. This is the first time the inverse scattering method was shown to be applicable to a lattice model]

B. Random Matrices

(1) Survey Articles

1. Random matrix models and their applications, edited by P. Bleher and A. Its, MSRI Publications, 40, Cambridge University Press, 2001. [This proceeding contains several survey papers by both mathematicians and physicists about various aspects of random matrix theory and its applications such as combinatorics, universality, solvable physical models and enumeration of Fyenman graphs.]

2. Journal of Physics A, Volume 36, Number 12, 2003. Special issue: Random Matrix Theory, edited by P. J. Forrester, N. C. Snaith and J. J. M. Verbaarschot. [This special issue contains survey and original papers for many different aspects of random matrix theory. Namely applications to number theory, applications to statistical mechanics, integrable systems, mesoscopic physics and disordered systems, quantum chaos and non-Hermtian matrices are discussed. Also contained is an article by the editors reviewing the main historical developments in random matrix theory.]

3. W. Kšnig, Orthogonal polynomial ensembles in probability theory, Probability Surveys, Volume 2, 385--447, 2005. [This paper surveys a number of models from physics, statistical mechanics, probability theory and combinatorics, which turned out to be described in terms of random matrix theory. This paper is aimed at non-experts with background in probability who want a quick survey of the field.]

4. P. Deift, Integrable systems and combinatorial theory, Notices of AMS, 47(6), 631--640, 2000. [This paper surveys the connections between increasing subsequences in combinatorics, random matrix theory and integrable systems.]

5. N. Katz and P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36(1), 1--26, 1999. [This is an expository paper concerning the connections between the zeros of zeta functions and random matrix theory.]

(2) Texts

1. M. L. Mehta, "Random Matrices", Academic Press. [This is a classic book on random matrices. The book contains many models and results from a physicist's view point.]

2. P. A. Deift, "Orthogonal Polynomials and Random Matrices", AMS. [This lecture note focuses on the orthogonal polynomial approach for universality question in Hermitian random matrices. A detailed account of Riemann-Hilbert analysis is presented.]

3. P. Forrester, Log-gases and Random matrices, available at http://www.ms.unimelb.edu.au/~matpjf/matpjf.html [This book in progress covers more recent topics that are not in the Mehta's book. Also one can find more details of proofs.]

4 N. Katz and P. Sarnak, "Random Matrices, Frobenius Eigenvalues, and Monodromy", AMS Colloquium Publications.

(3) Papers

1. C. A. Tracy and H. Widom, "Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices", J. Stat. Phys., volume 92, pp 809-835 (1998). [This paper presents an elementary proof of some of most fundamental algebraic formulas for correlations functions and gap distributions in random matrix theory.]