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A Survey of Peter D. Lax’s Contributions to Mathematics

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Abstract

We discuss Peter D. Lax’s contributions to mathematics over a period of more than 60 years.

Research supported in part by the Research Council of Norway.

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References

  1. Albers, D.J., Alexanderson, G.L., Reid, C. (eds.): More Mathematical People. Hartcourt Brace Jovanovich, Boston (1990)

    MATH  Google Scholar 

  2. Belokolos, E.D., Bobenko, A.I., Enol’skii, V.Z., Its, A.R., Matveev, V.B.: Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  3. Biachini, S., Bressan, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2) 61, 223–342 (2005)

    Article  MathSciNet  Google Scholar 

  4. Bressan, A.: Hyperbolic Systems of Conservation Laws. Oxford Univ. Press, Oxford (2000)

    MATH  Google Scholar 

  5. Bullough, R.K., Caudrey, P.J.: Solitons and the Korteweg–de Vries equation: Integrable systems in 1834–1995. Acta Appl. Math. 39, 193–228 (1995)

    Article  MathSciNet  Google Scholar 

  6. Chen, G.-Q.: Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics. III. Acta Math. Sci. 6, 75–120 (1986)

    Article  MathSciNet  Google Scholar 

  7. Chen, G.-Q., Liu, J.-G.: Convergence of difference schemes with high resolution for conservation laws. Math. Comput. 66, 1027–1053 (1997)

    Article  MathSciNet  Google Scholar 

  8. Colin de Verdiére, Y.: Pseudo-Laplacians. Ann. Inst. Fourier (Grenoble) 32, 275–286 (1982)

    Article  MathSciNet  Google Scholar 

  9. Conway, E., Smoller, J.: Global solutions of the Cauchy problem for quasilinear first-order equations in several space variables. Commun. Pure Appl. Math. 19, 95–105 (1966)

    Article  Google Scholar 

  10. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, New York (1976). (First published in 1948)

    Book  Google Scholar 

  11. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  12. Deift, P., McLaughlin, K.T.-R.: A continuum limit of the Toda lattice. Mem. Am. Math. Soc. 624, x+216 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Deift, P., Venakides, S., Zhao, X.: New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems. Internat. Math. Res. Notices 1997, 286–299 (1997)

    Article  MathSciNet  Google Scholar 

  14. Deift, P., Zhao, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. (2) 137, 294–368 (1993)

    Article  MathSciNet  Google Scholar 

  15. Delsarte, J.: Sur le gitter fuchsien. C. R. Acad. Sci. Paris 214, 147–179 (1942)

    MathSciNet  MATH  Google Scholar 

  16. Ding, X.X., Chen, G.-Q., Luo, P.Z.: Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics. I. Acta Math. Sci. 5, 415–432 (1985)

    Article  MathSciNet  Google Scholar 

  17. Ding, X.X., Chen, G.-Q., Luo, P.Z.: Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics. II. Acta Math. Sci. 5, 433–472 (1985)

    Article  MathSciNet  Google Scholar 

  18. Ding, X.X., Chen, G.-Q., Luo, P.Z.: A supplement to the papers: “Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics. II, III”. Acta Math. Sci. 9, 43–44 (1989)

    Article  Google Scholar 

  19. Dubrovin, B.A.: Inverse problem for periodic finite-zoned potentials in the theory of scattering. Funct. Anal. Appl. 9, 61–62 (1975)

    Article  Google Scholar 

  20. Dubrovin, B.A.: Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials. Funct. Anal. Appl. 9, 215–223 (1975)

    Article  Google Scholar 

  21. Dubrovin, B.A., Novikov, S.P.: A periodicity problem for the Korteweg–de Vries and Sturm–Liouville equations. Their connection with algebraic geometry. Dokl. Akad. Nauk SSSR 15, 1597–1601 (1974)

    MATH  Google Scholar 

  22. Dubrovin, B.A., Novikov, S.P.: Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg–de Vries equation. Sov. Phys. JETP 40, 1058–1063 (1975)

    MathSciNet  Google Scholar 

  23. Duistermaat, J.J.: Fourier Integral Operators. Birkhäuser, Basel (1996)

    MATH  Google Scholar 

  24. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39–79 (1975)

    Article  MathSciNet  Google Scholar 

  25. Duke, W., Rudnick, Z., Sarnak, P.: Density of integer points on affine homogeneous varieties. Duke Math. J. 71, 143–179 (1993)

    Article  MathSciNet  Google Scholar 

  26. Evans, L.C.: Partial Differential Equations. Am. Math. Soc., Providence (1998)

    MATH  Google Scholar 

  27. Faddeev, L., Pavlov, B.: Scattering theory and automorphic functions. Proc. Steklov Inst. Math. 27, 161–193 (1972)

    MathSciNet  Google Scholar 

  28. Gorodnik, A., Oh, H., Shah, N.: Integral points on symmetric varieties and Satake compatifications: Preprint, arXiv:math/0610497v2 (2008)

  29. Gardner, C.S.: Korteweg–de Vries equation and generalizations. IV. The Korteweg–de Vries equation as a Hamiltonian system. J. Math. Phys. 12, 1548–1551 (1971)

    Article  MathSciNet  Google Scholar 

  30. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)

    Article  Google Scholar 

  31. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Korteweg–de Vries equation and generalizations. VI. Methods for exact solution. Commun. Pure Appl. Math. 27, 97–133 (1974)

    Article  Google Scholar 

  32. Gel’fand, I.M.: Some problems in the theory of quasilinear equations. Am. Math. Soc. Transl. 29, 295–381 (1963)

    MathSciNet  MATH  Google Scholar 

  33. Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions. (1+1)-Dimensional Continuous Models, vol. I. Cambridge Univ. Press, Cambridge (2003)

    Book  Google Scholar 

  34. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)

    Article  MathSciNet  Google Scholar 

  35. Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws. Springer, New York (2007). Corrected 2nd printing

    MATH  Google Scholar 

  36. Hopf, E.: The partial differential equation u t+uu x=μu xx. Commun. Pure Appl. Math. 3, 201–230 (1950)

    Article  Google Scholar 

  37. Hörmander, L.: Pseudo-differential operators and non-elliptic boundary value problems. Ann. Math. 83, 129–209 (1966)

    Article  MathSciNet  Google Scholar 

  38. Hörmander, L.: The spectrum of a positive elliptic operator. Acta Math. 121, 193–218 (1968)

    Article  MathSciNet  Google Scholar 

  39. Hörmander, L.: Fourier integral operators. I. Acta Math. 127, 79–183 (1971)

    Article  MathSciNet  Google Scholar 

  40. Ikawa, M.: On the poles of the scattering matrix for two strictly convex obstacles. J. Math. Kyoto Univ. 23, 127–194 (1983). Addendum, loc. sit. 23, 795–802 (1983)

    Article  MathSciNet  Google Scholar 

  41. Its, A.R., Matveev, V.B.: Hill’s operator with finitely many gaps. Funct. Anal. Appl. 9, 65–66 (1975)

    Article  Google Scholar 

  42. Its, A.R., Matveev, V.B.: Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg–de Vries equation. Theoret. Math. Phys. 23, 343–355 (1975)

    Article  Google Scholar 

  43. Jin, S., Levermore, C.D., McLaughlin, D.W.: The behavior of solutions of the NLS equation in the semiclassical limit. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds.) Singular Limits of Dispersive Waves. NATO Adv. Sci. Inst. Ser. B, Phys., vol. 320, pp. 235–255. Plenum, New York (1994)

    Chapter  Google Scholar 

  44. Klainerman, S., Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Commun. Pure Appl. Math. 33, 241–263 (1980)

    Article  MathSciNet  Google Scholar 

  45. Kružkov, S.N.: First order quasi-linear equations in several independent variables. Math. USSR Sb. 10, 217–243 (1970)

    Article  Google Scholar 

  46. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Univ. Press, Cambridge (2002)

    Book  Google Scholar 

  47. Ludwig, D., Morawetz, C.: The generalized Huygens’ principle for reflecting bodies. Commun. Pure Appl. Math. 22, 189–205 (1969)

    Article  Google Scholar 

  48. Melrose, R.: Singularities and energy decay in acoustical scattering. Duke Math. J. 46, 43–59 (1979)

    Article  MathSciNet  Google Scholar 

  49. Melrose, R.: Polynomial bound on the number of scattering poles. J. Funct. Anal. 53, 287–303 (1983)

    Article  MathSciNet  Google Scholar 

  50. McKean, H.P., van Moerbeke, P.: The spectrum of Hill’s equation. Invent. Math. 30, 217–274 (1975)

    Article  MathSciNet  Google Scholar 

  51. Müller, W.: The trace class conjecture in the theory of automorphic forms. Ann. Math. 130, 473–529 (1989)

    Article  MathSciNet  Google Scholar 

  52. Novikov, S.P.: The periodic problem for the Korteweg–de Vries equation. Funct. Anal. Appl. 8, 236–246 (1974)

    Article  MathSciNet  Google Scholar 

  53. Novikov, S.P.: A method for solving the periodic problem for the KdV equation and its generalizations. Rocky Mountain J. Math. 8, 83–93 (1978)

    Article  MathSciNet  Google Scholar 

  54. Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons. Consultants Bureau, New York (1984)

    MATH  Google Scholar 

  55. Oleĭnik, O.A.: Discontinuous solutions of non-linear differential equations. Am. Math. Soc. Transl. Ser. 26, 95–172 (1963)

    MathSciNet  MATH  Google Scholar 

  56. Oleĭnik, O.A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Am. Math. Soc. Transl. Ser. 33, 285–290 (1963)

    MATH  Google Scholar 

  57. Patterson, S.: The Laplacian operator on a Riemann surface. I. Compos. Math. 32, 83–107 (1975)

    MathSciNet  MATH  Google Scholar 

  58. Patterson, S.: The Laplacian operator on a Riemann surface. II. Compos. Math. 32, 71–112 (1976)

    MathSciNet  MATH  Google Scholar 

  59. Patterson, S.: The Laplacian operator on a Riemann surface. III. Compos. Math. 33, 227–259 (1976)

    MathSciNet  MATH  Google Scholar 

  60. Patterson, S.J., Perry, P.A.: The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J. 106, 321–390 (2001)

    Article  MathSciNet  Google Scholar 

  61. Pego, R.: Origin of the KdV equation. Not. Am. Math. Soc. 45, 358 (1998)

    Google Scholar 

  62. Phillips, R., Sarnak, P.: Perturbation theory for the Laplacian on automorphic functions. J. Am. Math. Soc. 5, 1–32 (1992)

    Article  MathSciNet  Google Scholar 

  63. Ralston, J.: Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22, 807–823 (1969)

    Article  MathSciNet  Google Scholar 

  64. Riemann, G.F.B.: Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abh. König. Gesell. Wiss. Göttingen 8, 43–65 (1860)

    Google Scholar 

  65. Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956)

    MathSciNet  MATH  Google Scholar 

  66. Serre, D.: Systems of Conservation Laws, vol. 1. Cambridge Univ. Press, Cambridge (1999)

    Book  Google Scholar 

  67. Smoller, J.: Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer, New York (1994)

    Book  Google Scholar 

  68. Sogge, C., Zelditch, S.: Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114, 387–437 (2002)

    Article  MathSciNet  Google Scholar 

  69. Venakides, S.: The zero dispersion limit of the Korteweg–de Vries equation for initials with nontrivial reflection coefficient. Commun. Pure Appl. Math. 38, 125–155 (1985)

    Article  MathSciNet  Google Scholar 

  70. Venakides, S.: The zero dispersion limit of the Korteweg–de Vries equation for initials with periodic initial data. Trans. Am. Math. Soc. 301, 189–226 (1987)

    Article  MathSciNet  Google Scholar 

  71. Venakides, S.: The continuum limit of theta functions. Commun. Pure Appl. Math. 42, 711–728 (1989)

    Article  MathSciNet  Google Scholar 

  72. Venakides, S., Deift, P., Oba, R.: The Toda shock problem. Commun. Pure Appl. Math. 44, 1171–1242 (1991)

    Article  MathSciNet  Google Scholar 

  73. Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)

    Article  Google Scholar 

  74. Zakharov, V.E., Faddeev, L.D.: Korteweg–de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl. 5, 280–287 (1971)

    Article  Google Scholar 

  75. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)

    MathSciNet  Google Scholar 

  76. Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl. 8, 226–235 (1974)

    Article  Google Scholar 

  77. Zakharov, V.E., Shabat, A.B.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl. 13, 166–174 (1979)

    Article  MathSciNet  Google Scholar 

  78. Zworski, M.: Sharp polynomial bounds on the number of scattering poles for radial potentials. J. Funct. Anal. 82, 370–403 (1989)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Helge Holden

  2. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Helge Holden

  3. School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Peter Sarnak

  4. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Peter Sarnak

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  1. Helge Holden
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  2. Peter Sarnak
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Correspondence to Helge Holden .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Helge Holden

  2. Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, 0316, Oslo, Norway

    Ragni Piene

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Holden, H., Sarnak, P. (2010). A Survey of Peter D. Lax’s Contributions to Mathematics. In: Holden, H., Piene, R. (eds) The Abel Prize. The Abel Prize. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01373-7_9

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