Henry P. McKean Jr. Selecta pp 15-30 | Cite as

# Henry P. McKean Jr. and Integrable Systems

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## Abstract

Henry McKean’s first contribution to integrable systems appeared in 1975 [15]. Over a period of more than 30 years since, in some 50 papers, he has explored integrable systems from uniquely original points of view. His selecta could have included the pioneering work, with Airault and Moser, on the time-dynamics of poles of meromorphic solutions of KdV [2]; or the series on invariant measures for wave equations, integrable or otherwise; or one of the seminal papers with Trubowitz [12, 13] that created a theory of infinite-genus hyperelliptic curves (= Riemann surfaces) and infinite-dimensionalJacobian varieties, which they applied to KdV under periodic boundary conditions, or subsequent extensions of that theory motivated by the desire to understand all the iconic integrable partial differential equation on the circle.

## Keywords

Vector Field Riemann Surface Theta Function Inverse Scattering Darboux Transformation## Bibliography

- [1]M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The inverse scattering transform. Fourier analysis for nonlinear problems.
*Stud. Appl. Math.*, 53:249–315, 1974.Google Scholar - [2]H. Airault, H. P. McKean, and J. Moser. Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem.
*Comm. Pure Appl. Math.*, 30:95–148, 1977.MathSciNetCrossRefzbMATHGoogle Scholar - [3]N. Ercolani and H. P. McKean. Geometry of KdV (4). Abel sums, Jacobi variety, and theta function in the scattering case.
*Invent. Math.*, 99(3):483–544, 1990.Google Scholar - [4]C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Method for solving the Korteweg-de Vries equation.
*Phys. Rev. Letters*, 19:1095–1097, 1967.CrossRefGoogle Scholar - [5]A. R. Its and V. B. Matveev. Hill operators with a finite number of lacunae.
*Funkcional. Anal. i Priložen.*, 9(1):69–70, 1975.MathSciNetCrossRefGoogle Scholar - [6]V. B. Matveev.
*Darboux Transformations in Nonlinear Dynamics*. Springer Series in Nonlinear Dynamics. Springer-Verlag, 1991.CrossRefGoogle Scholar - [7]H. P. McKean. Geometry of KdV (1): addition and the unimodal spectral class.
*Rev. Math. Iberoamer.*, 2:235–261, 1986.MathSciNetCrossRefzbMATHGoogle Scholar - [8]H. P. McKean. Geometry of KdV (2): three examples.
*J. Stat. Phys.*, 46:1115–1143, 1987.MathSciNetCrossRefzbMATHGoogle Scholar - [9]H. P. McKean. Is there an Infinite-Dimensional Algebraic Geometry? Hints from KDV. In L. Ehrenpreis and R. C. Gunning, editors,
*Proceedings of Symposia in Pure Mathematics*, volume 49(1), pages 27–37. American Mathematical Society, 1989.Google Scholar - [10]H. P. McKean. Geometry of KdV (3): Determinants and Unimodular Isospectral Flows.
*Comm. Pure Appl. Math.*, 45:389–415, 1992.MathSciNetCrossRefzbMATHGoogle Scholar - [11]H. P. McKean. Book reviews:
*Riemann surfaces of infinite genus*by J. Feldman, H. Knorrer and E. Trubowitz.*Bull. Amer. Math. Soc.*, 42:79–87, 2005.Google Scholar - [12]H. P. McKean and E. Trubowitz. Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points.
*Comm. Pure Appl. Math.*, 29:143–226, 1976.MathSciNetCrossRefzbMATHGoogle Scholar - [13]H. P. McKean and E. Trubowitz. Hill surfaces and their theta functions.
*Bull. Amer. Math. Soc.*, 84:1042–1085, 1978.MathSciNetCrossRefzbMATHGoogle Scholar - [14]H. P. McKean and E. Trubowitz. The spectral class of the quantum-mechanical harmonic oscillator.
*Comm. Math. Phys.*, 82(4):471–495, 1981/82.Google Scholar - [15]H. P. McKean and P. van Moerbeke. The spectrum of Hill’s equation.
*Inv. Math.*, 30:217–274, 1975.CrossRefzbMATHGoogle Scholar - [16]V. E. Zakharov and L. D. Faddeev. The Korteweg-de Vries equation: a completely integrable Hamiltonian system.
*Funct. Anal. Applic.*, 5:280–287, 1971.CrossRefGoogle Scholar