REFERENCES
Βasic Reference
• P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach, AMS, 2000.
Α. Classical Theory of Random Matrices
• E.Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Annals of Mathematics 62(3), 1955, pp. 548–564;
Characteristic vectors of bordered matrices of infinite dimensions II. Ann. Math. v.65 (1957) no.2, 203-207;
On the distribution of the roots of certain symmetric matrices. Ann. Math. v.67 (1958) no.2, 325-326
• F. J. Dyson, Statistical Theory of the Energy Levels of Complex Systems, I-III, J. Math. Physics 3 (1962)
• F. J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math.Physics 3 (1962), pp. 1191–1198
J. Math.Physics 3 (1962), p.1199
• F. J. Dyson, M.L. Mehta, Statistical Theory of the Energy Levels of Complex Systems. IV-V,J. Math. Physics 4 (1963)
• F. J. Dyson, Correlations between eigenvalues of a random matrix, Comm. Math. Phys. 19 (1970), no. 3, 235-250.
• F. J. Dyson, Fredholm Determinants and Inverse Scattering Problems, Comm. Math. Phys. 47 (1976), no. 3, 171-183.
• M.L. Mehta, Random Matrices, Academic Press,1967; third edition: Elsevier 2004.
Β. Basic Theory of Cauchy and Hilbert Operators and Riemann-Hilbert problems
• F. D. Gakhov, Boundary Value Problems, Dover, New York, 1990 (reprint of 1966 edition).
• Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.
• K. Clancey, I. Gokhberg, Factorization of Matrix Functions and Singular Integral Operators (Birkhäuser, Basel, 1981)
• R. Beals, R.Coifman, Scattering and inverse scattering for first order systems, Communications in Pure and Applied Mathematics, pp.39-90, v.37 (1984)
• R. Beals, P. Deift, C. Tomei, Direct and Inverse Scattering on the Line, AMS 1988.
• X.Zhou, Direct and inverse scattering transforms with arbitrary spectral singularities, Communications on Pure and Applied Mathematics, vol. 42, no. 7, Jan. 1989, pp. 895–938.
• P. Deift, X. Zhou, Direct and inverse scattering on the line with arbitrary singularities, Communications on Pure and Applied Mathematics, vol. 44, no. 5, Jan. 1991, pp. 485–533.
• X,Zhou, L2-Sobolev space bijectivity of the scattering and inverse scattering transforms, Communications on Pure and Applied Mathematics, vol. 51, no. 7, Jan. 1998.
C. Applications of RHP to ODEs & PDEs, orthogonal polynomials and random matrices
• A. R. Its, Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations, Doklady Akad. Nauk S. S. S. R. 261(1), 14–18 (1982)
• A.R.Its, V.Yu.Novokshenov, “The Isomonodromic Deformation Method in the Theory of Painlevé Equations,” Lecture Notes in Math., 1191, Springer-Verlag, 1986.
• A.S.Fokas, A.R.Its, A.V.Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), no. 2, 395--430
• A.S.Fokas, X. Zhou, On the solvability of Painlevé II and IV, Communications in Mathematical Physics, vol. 144, n. 3,1992, pp. 601–22.
• A.S.Fokas, U.Mugan, X. Zhou, On the solvability of Painlevé I, III and V, Inverse Problems, vol. 8, no. 5, Dec. 1992, pp. 757–85
• P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems; Asymptotics for the MKdV equation, Annals of Mathematics, 137(2), 295-368 (1993).
• P. Deift, S. Venakides, X. Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Communications in Pure and Applied Mathematics, 47, 199-206 (1994).
• P. Deift, A. R. Its, and X. Zhou, A Riemann-Hilbert Approach to Asymptotic Problems Arising in the Theory of Random Matrix Models, and also in the Theory of Integrable Statistical Mechanics, Ann.Math, Vol. 146, No. 1 (Jul., 1997), pp. 149-235
• A.S.Fokas, A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, Painlevé Transcendents: The Riemann–Hilbert Approach (AMS, 2006)
•Spyridon Kamvissis, Gerald Teschl, Long Time Asymptotics of the Periodic Toda Lattice under Short Range Perturbations, J. Math. Physics, v.53, n.7, 2012
• Y.Rybalko, D.Shepelsky, Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation, J.Math.Physics, 60, 031504, 2019
D. Applications to integrable probability
• J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, Journal of the American Mathematical Society, 12(4), 1119-1178 (1999).
• Mattia Cafasso, Tom Claeys, A Riemann-Hilbert approach to the lower tail of the KPZ equation, arXiv:1910.02493
Ε. Semiclassical problems and auxiliary variational problems of electrostatic type
• Peter D. Lax, C. David Levermore, The small dispersion limit of the Korteweg‐de Vries equation: parts I, II, III,
Communications in Pure and Applied Mathematics, v.36 (1983).
• P. Deift, S. Venakides, X. Zhou, An extension of the steepest descent method for Riemann-Hilbert problems: the small dispersion limit of the Korteweg-de Vries (KdV) equation, PNAS, Jan 1998.
•S. Kamvissis, E. A. Rakhmanov, Existence and Regularity for an Energy Maximization Problem in Two Dimensions, J. Math. Physics, v.46, n.8, 2005
F. IBVPs. Unified Transform Method.
• A. S. Fokas, On the integrability of linear and nonlinear partial differential equations, J. Math. Physics 41 6 (2000), 4188-4237
• A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 1 (2002), 1-39
• A. S. Fokas, A. R. Its, L. Y. Sung The nonlinear Schrödinger equation on the half-line, Nonlinearity 18 4 (2005), 1771-1822.
G. Review papers
• P. Deift, A. R. Its, and X. Zhou, “Long-time asymptotics for integrable nonlinear wave equations,” in Important Developments in Soliton Theory 1980-1990, edited by A. S. Fokas and V. E. Zakharov (Springer, New York, 1993), pp. 181–204.
• A. R. Its The Riemann-Hilbert problem and integrable systems, Notices Amer. Math. Soc. 50 11 (2003), 1389-1400
• S.Kamvissis, From Stationary Phase to Steepest Descent, Contemporary Mathematics 2007.