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Three Lectures on “Fifty Years of KdV: An Integrable System”

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Nonlinear Dispersive Partial Differential Equations and Inverse Scattering

Part of the book series: Fields Institute Communications ((FIC,volume 83))

Abstract

The goal in the first two Coxeter lectures was to give an answer to the question

$$\displaystyle \mbox{``What is an integrable system?''} $$

and to describe some of the tools that are available to identify and integrate such systems. The goal of the third lecture was to describe the role of integrable systems in certain numerical computations, particularly the computation of the eigenvalues of a random matrix. This paper closely follows these three Coxeter lectures, and is written in an informal style with an abbreviated list of references. Detailed and more extensive references are readily available on the web. The list of authors mentioned is not meant in any way to be a detailed historical account of the development of the field and I ask the reader for his’r indulgence on this score.

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Notes

  1. 1.

    This paper also contains some history of earlier approaches to analyze the behavior of solutions of integrable systems asymptotically.

References

  1. J. Baik, P. Deift and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12, 1999, No. 4, 1119–1178.

    Article  MathSciNet  Google Scholar 

  2. J. Baik, P. Deift and T. Suidan, Combinatorics and random matrix theory, Graduate Studies in Mathematics 172, American Mathematical Society, Providence, 2016.

    Book  MATH  Google Scholar 

  3. Y. Bakhtin and J. Correll, A neural computation model for decision-making times, J. Math. Psychol. 56(5), 2012, 333–340.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Beals and R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37, 1984, 39–90.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Deift, L-C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math. 39, 1986, 183–232.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Deift, L-C. Li and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math. 42, 1989, 443–521.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Deift, G. Menon, S. Olver and T. Trogdon, Universality in numerical computations with random data, Proc. Natl. Acad. Sci. USA 111, 2014, no. 42, 14973–14978.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Deift, G. Menon and C. Pfrang, How long does it take to compute the eigenvalues of a random symmetric matrix?, Random matrix theory, interacting particle systems, and integrable systems, 411–442, Math. Sci. Res. Inst. Publ. 65, Cambridge Univ. Press, New York, 2014.

    Google Scholar 

  9. P. Deift, T. Nanda and C. Tomei, Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal. 20, No. 1, 1983, 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, Intl. Math. Res. Not. IMRN 2011, no. 24, 5505–5624.

    Google Scholar 

  11. P. Deift and T. Trogdon, Universality for the Toda algorithm to compute the largest eigenvalue of a random matrix, Comm. Pure Appl. Math. 71, 2018, no. 3, 505–536.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Deift and T. Trogdon, Universality for eigenvalue algorithms on sample covariance matrices, SIAM J. Numer. Anal. 55, No. 6, 2017, 2835–2862.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. Second Series 137, No. 2 1993, 295–368.

    Google Scholar 

  14. P. Deift and X. Zhou, Long-time asymptotics for the autocorrelation function of the transverse Ising chain at the critical magnetic field, Singular limits of dispersive waves (Lyon, 1991), 183–201, NATO Adv. Sci. Inst. (Series B:Physics), 320, Plenum, New York, 1994.

    Google Scholar 

  15. P. Deift and X. Zhou, Perturbation theory for infinite-dimensional integrable systems on the line. A case study, Acta Math. 188, No. 2, 2002, 163–262.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Erdős and H. T. Yau, A dynamical approach to random matrix theory, Courant Lecture Notes 28, American Math. Soc., Providence, 2017.

    Google Scholar 

  17. H. Flaschka, The Toda lattice, I, Phys. Rev. B 9, 1974, 1924–1925.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Fokas, A. Its, A. Kitaev and V. Novokshenov, Painlevé transcendents: The Riemann-Hilbert approach, Math. Surveys and Monographs, Vol. 128, Amer. Math. Soc. Providence, 2006

    Google Scholar 

  19. C. Garner, J. Greene, M. Kruskal and R. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett. 19, 1967, 1095–1097.

    Article  MATH  Google Scholar 

  20. P. Gavrylenko and O. Lisovyy, Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions, Comm. Math. Phys. 363, No. 1, 2018, 1–58.

    Google Scholar 

  21. P. Gérard and E. Lenzman, A Lax pair structure for the half-wave maps equation, Lett. Math. Phys. 108, No. 7, 2018, 1635–1648.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Holmer and M. Zworski, Breathing patterns in nonlinear relaxation, Nonlinearity, 22(6), 2009, 1259–1301.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Its, A. Izergin, V. Korepin and N. Slavnov, Differential equations for quantum correlation functions, Int. J. Mod. Physics, B4, 1990, 1003–1037.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Its and D. Shepelsky, Initial boundary value problem for the focusing nonlinear Schrödinger equation with Robin boundary condition: half-line approach, Proc. R. Soc. Lond. Ser A, Math. Phys. Eng. Sci. 469 2013, 2149, 20120199, 14 pages.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Jimbo, T. Miwa, Y. Môri and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D, Vol. 1, No. 1, 1980, 80–158.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21(5), 1968, 467–490.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Manakov, Complete integrability and stochastization of discrete dynamical systems, Sov. Phys. JETP 40, 1974, 269–274.

    MathSciNet  Google Scholar 

  28. B. McCoy, J. Perk and R. Schrock, Time-dependent correlation functions of the transverse Ising chain at the critical magnetic field, Nucl. Phys. B220, 1983, 35–47.

    Article  MathSciNet  Google Scholar 

  29. J. Moser, Finitely many mass points on the line under the influence of an exponential potential—An integrable system, Dynamical Systems Theory and Applications, J. Moser, ed., Springer-Verlag, New York, Berlin, Heidelberg, 1975, 467–497.

    Google Scholar 

  30. A.B. Shabat, One-dimensional perturbations of a differential operator and the inverse scattering problem, in Problems in Mechanics and Mathematical Physics, Nauka, Moscow, 1976.

    Google Scholar 

  31. W. Symes, TheQRalgorithm and scattering for the finite nonperiodic Toda lattice, Physica D, Vol. 4, No. 2, 1982, 275–280.

    Article  MathSciNet  MATH  Google Scholar 

  32. V. Zakharov and L. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funktsional. Anal. i Prilozhen, 5:4, 1971, 18–27.

    Google Scholar 

  33. V. Zakharov and S. Manakov, On the complete integrability of a nonlinear Schrödinger equation, Journal of Theoretical and Mathematical Physics, 19(3), 1974, 551–559.

    Article  MATH  Google Scholar 

  34. V. Zakharov and S. Manakov, Asymptotic behavior of nonlinear wave systems integrated by the inverse method, Zh. Eksp. Teor. Fiz., 71, 1976, 203–215; Soviet Physics JETP 44, No. 1, 1976, 106–112.

    Google Scholar 

  35. V. Zakharov and A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Zh. Eksp. Teor. Fiz. 61, 1971, 118–134; Soviet Physics JETP 34, 1972, 62–69.

    Google Scholar 

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Acknowledgements

The work of the author was supported in part by NSF Grant DMS–1300965.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Percy A. Deift

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  1. Percy A. Deift
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Correspondence to Percy A. Deift .

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Editors and Affiliations

  1. Department of Mathematics, University of Michigan‑Ann Arbor, Ann Arbor, MI, USA

    Peter D. Miller

  2. Department of Mathematics, University of Kentucky, Lexington, KY, USA

    Peter A. Perry

  3. Département de Mathématiques, University of Paris-Sud, Orsay, France

    Jean-Claude Saut

  4. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Catherine Sulem

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Deift, P.A. (2019). Three Lectures on “Fifty Years of KdV: An Integrable System”. In: Miller, P., Perry, P., Saut, JC., Sulem, C. (eds) Nonlinear Dispersive Partial Differential Equations and Inverse Scattering. Fields Institute Communications, vol 83. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9806-7_1

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