Skip to main content
SpringerLink
Log in

On the ubiquitous Gelfand-Levitan-Marčenko (GLM) equation

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

We collect a number of derivations and interpretations of Gelfand-Levitan-Marčenko (GLM) equations in various contexts. Relations to Riemann-Hilbert (RH) problems, tau functions and determinants, direct linearization techniques, etc. are discussed and some new results are included. We emphasize the role of spectral pairings of generalized eigenfunctions and exhibit various ways in which inverse scattering techniques arise. The natural extension and generalization of GLM techniques in the format of RH methods and coadjoint orbits in a Lie theoretic context is developed and the hierarchy point of view is explored in terms of spectral ideas. Thus, the roles and some limitations of the GLM and RH methods are exposed to some extent and some bridges are sketched to modern techniques in soliton mathematics involving hierarchies, tau functions, Lie theory, etc.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ablowitz, M., Bar Yaacov, D., and Fokas, A.: Stud. Appl. Math. 69 (1983), 135–143.

    Google Scholar 

  2. Ablowitz, M. and Nachman, A.: Physica 18D (1986), 223–241.

    Google Scholar 

  3. Ablowitz, M. and Segur, H.: Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.

    Google Scholar 

  4. Ablowitz, M. and Fokas, A.: Lecture Notes in Phys. 189, Springer, New York, 1983, pp. 3–24.

    Google Scholar 

  5. Ablowitz, M., Kaup, D., Newell, A., and Segur, H.: Stud. Appl. Math. 53 (1974), 249–315.

    Google Scholar 

  6. Adams, M., Ratiu, T., and Schmid, R.: Infinite Dimensional Groups with Applications, Springer, New York, 1985, pp. 1–70.

    Google Scholar 

  7. Adams, M., Harnad, J., and Previato, E.: Comm. Math. Phys. 117 (1988), 451–500.

    Google Scholar 

  8. Adler, M.: Invent. Math. 50 (1979), 219–248.

    Google Scholar 

  9. Adler, M. and Moser, J., Comm. Math. Phys. 61 (1978), 1–30.

    Google Scholar 

  10. Adler, M., and Van Moerbeke, P.: Adv. in Math. 38 (1980), 267–317, 318–379.

    Google Scholar 

  11. Athorne, C. and Fordy, A.: J. Math. Phys. 28 (1987), 2018–2024.

    Google Scholar 

  12. Atiyah, M. and Hitchin, N.: The Geometry and Dynamics of Magnetic Monopoles, Princeton Univ. Press, 1988.

  13. Beals, R. and Coifman, R.: Comm. Pure Appl. Math. 37 (1984), 39–90; 38 (1985), 29–42.

    Google Scholar 

  14. Beals, R. and Coifman, R.: Proc. Symp. Pure Math. 43 (1985), 45–70.

    Google Scholar 

  15. Beals, R. and Coifman, R.: Physica 18D (1986), 242–249; Inverse Problems 5 (1989), 87–130.

    Google Scholar 

  16. Beals, R., Deift, P., and Tomei, C.: Direct and Inverse Scattering on the Line, Amer. Math. Soc., Providence, RI, 1988.

    Google Scholar 

  17. Bergvelt, M. and ten Kroode, A.: Lett. Math. Phys. 12 (1986), 139–147; J. Math. Phys. 28 (1987), 302–306; 29 (1988), 1308–1320.

    Google Scholar 

  18. Bruckstein, A., Kailath, T., and Levy, B.: SIAM J. Appl. Math. 45 (1985), 312–335.

    Google Scholar 

  19. Burridge, R.: Wave Motion 2 (1980), 305–323.

    Google Scholar 

  20. Calogero, F. and Degasperis, A.: Spectral Transform and Solitons, I, North Holland, Amsterdam, 1982.

    Google Scholar 

  21. Carroll, R.: Transmutation Theory and Applications, North-Holland, Amsterdam, 1985.

    Google Scholar 

  22. Carroll, R.: Mathematical Physics, North-Holland, Amsterdam, 1988.

    Google Scholar 

  23. Carroll, R.: Proc. Japan Acad. Series A Math. Sci. 60 (1984), 82–85.

    Google Scholar 

  24. Carroll, R.: CR Royal Soc. Canada 9 (1987), 237–242; Appl. Anal. 26 (1987), 61–85.

    Google Scholar 

  25. Carroll, R.: Appl. Anal. 18 (1984), 39–54; with S. Doizycki, Appl. Anal. 19 (1985), 189–200.

    Google Scholar 

  26. Carroll, R.: Oakland Conf. PDE, Pitman Press, 1987, pp. 1–38.

  27. Carroll, R.: Appl. Anal. 22 (1986), 21–43.

    Google Scholar 

  28. Carroll, R.: Abstract Methods in PDE, Harper Row, New York, 1969.

    Google Scholar 

  29. Carroll, R.: Boll. Un. Mat. Ital. 5B (1986), 465–486.

    Google Scholar 

  30. Carroll, R.: Some remarks on KP, Proc. L. K. Hua Sympos., Beijing, 1988, to appear.

  31. Carroll, R. and Dolzycki, S.: J. Math. Phys. 25 (1984), 91–93.

    Google Scholar 

  32. Carroll, R.: Appl. Anal. 32 (1989), 177–182.

    Google Scholar 

  33. Carroll, R. and Santosa, F.: J. Acoust. Soc. Amer. 76 (1984), 935–941.

    Google Scholar 

  34. Carroll, R. and Delic, G.: Math. Methods Appl. Sci. 11 (1989), 43–63.

    Google Scholar 

  35. Carroll, R.: Transmutation, scattering Theory, and Special Functions, North-Holland, Amsterdam, 1982.

    Google Scholar 

  36. Carroll, R.: Some remarks on the Hirota equations, to appear.

  37. Carroll, R.: On some relations involving tau functions, dressing kernels, and spectral data, to appear.

  38. Carroll, R.: Some connections between inverse scattering and soliton hierarchies, in preparation.

  39. Chadan, K. and Sabatier, P.: Inverse Problems in Quantum Scattering Theory, Springer, New York, 1977.

    Google Scholar 

  40. Date, E., Kashiwara, M., Jimbo, M., and Miwa, T.: RIMS Sympos., World Scientific, Singapore, 1983, pp. 39–119.

    Google Scholar 

  41. Date, E., Kashiwara, M., Jimbo, M., and Miwa, T.: Physica 4D (1982), 343–365.

    Google Scholar 

  42. Date, E., Kashiwara, M., and Miwa, T.: Proc. Japan Acad. Series A. Math. Sci. 57 (1981), 387–392.

    Google Scholar 

  43. Deift, P. and Trubowitz, E.: Comm. Pure Appl. Math. 32 (1979), 121–251.

    Google Scholar 

  44. Dhooghe, P.: J. Geom. Phys. 1 (1984), 9–38.

    Google Scholar 

  45. Dorfmeister, J., Neher, E., and Szmigielski, J.: Quart. J. Math. Oxford Ser. 2 40 (1989), 161–195.

    Google Scholar 

  46. Drinfeld, V. and Sokolov, V.: Soviet Math. Dokl. 23 (1981), 457–462; Soviet Prob. Math. 24 (1984), 81–180.

    Google Scholar 

  47. Dym, H. and Iacob, A.: Topics in Operator Theory, Birkhauser, Basle, 1984, pp. 141–240; Toeplitz Centen., Birkhauser, Basle, 1982, pp. 233–260.

    Google Scholar 

  48. Dyson, F.: Stud. Math. Phys., Princeton Univ. Press, 1976, pp. 151–167.

  49. Fadeev, L. and Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons, Springer, New York, 1987.

    Google Scholar 

  50. Fadeev, L.: Uspekhi Mat. Nauk. 14 (1959), 57–119; Soviet Prob. Math. 31 (1974), 93–180.

    Google Scholar 

  51. Flaschka, H., Newell, A., and Ratiu, T.: Physica 9D (1983), 300–323; 324–332.

    Google Scholar 

  52. Flaschka, H.: Quart. J. Math. Oxford Ser. 2, 34 (1983), 61–65.

    Google Scholar 

  53. Fokas, A. and Ablowitz, M.: Stud. Appl. Math. 69 (1983), 211–228.

    Google Scholar 

  54. Fokas, A. and Ablowitz, M.: Lecture Notes in Phys. 189, Springer, New York, 1983, pp. 137–183.

    Google Scholar 

  55. Fokas, A. and Ablowitz, M.: Phys. Rev. Lett. 47 (1981), 1096–1100.

    Google Scholar 

  56. Fokas, A. and Ablowitz, M.: Phys. Lett. 94A (1983), 67–70.

    Google Scholar 

  57. Fomenko, A. and Trofimov, V.: Integrable Systems on Lie Algebras and Symmetric Spaces, Gordon and Breach, 1988.

  58. Fordy, A. and Kulish, P.: Comm. Math. Phys. 89 (1983), 427–443.

    Google Scholar 

  59. Gelfand, I. and Levitan, B.: Izv. Akad. Nauk SSSR 15 (1951), 309–360.

    Google Scholar 

  60. Gerdzikov, V.: Inverse Prob. 2 (1986), 51–74.

    Google Scholar 

  61. Gerdzikov, V. and Yanovski, A.: Comm. Math. Phys. 103 (1986), 549–568.

    Google Scholar 

  62. Gilbert, G.: Comm. Math. Phys. 111 (1988), 331–348.

    Google Scholar 

  63. Helminck, G.: J. Math. Phys. 29 (1983), 1974–1978; with G. Post, Lett. Math. Phys. 16 (1988), 359–364.

    Google Scholar 

  64. Jimbo, M. and Miwa, T.: Publ. RIMS, Kyoto Univ. 19 (1983), 943–1001.

    Google Scholar 

  65. Kac, V. and Raina, A.: Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, Singapore, 1987.

    Google Scholar 

  66. Kailath, T.: Stochastic Systems: The Mathematics of Filtering and Identification, D. Reidel, Dordrecht, 1981, pp. 307–350.

    Google Scholar 

  67. Kaku, M.: Introduction to Superstrings, Springer, New York, 1988.

    Google Scholar 

  68. Kato, Y.: Prog. Theoret. Phys. 76 (1986), 768–783; 78 (1987), 198–213.

    Google Scholar 

  69. Katsura, T., Shimizu, Y., and Ueno, K.: Comm. Math. Phys. 121 (1989), 603–627.

    Google Scholar 

  70. Kawamoto, N., Namikawa, Y., Tsuchiya, A., and Yamada, Y.: Comm. Math. Phys. 116 (1988), 247–308.

    Google Scholar 

  71. Kawata, T.: J. Phys. Soc. Japan, 51 (1982), 3381–3389; 53 (1984), 2847–2884; 54 (1985), 57–63 and 3708–3717; 56 (1987), 1261–1263; 57 (1988), 57–65 and 422–435.

    Google Scholar 

  72. Krichever, I.: Funct. Anal. Appl. 11 (1977), 12–26; 18 (1984), 210–223; Uspekhi Mat. Nauk. 32 (1977), 183–208; Soviet Prob. Math. 23 (1983), 79–136.

    Google Scholar 

  73. Kupershmidt, B.: Astérisque 123, 1985.

  74. Lee, J.: Chinese J. Math. 12 (1984), 223–233; 14 (1986), 225–248.

    Google Scholar 

  75. Levitan, B.: Inverse Sturm Liouville Problems, Moscow, 1984.

  76. Manakov, S.: Physica 3D (1981), 306–352.

    Google Scholar 

  77. Marcenko, V.: Sturn Liouville Operators and their Applications, Moscow, 1977; Nonlinear Equations and Operator Algebras, D. Reidel, Dordrecht, 1987.

  78. McKean, H.: PDE and Geometry, Dekker, 1979, pp. 237–254.

  79. McKean, H.: Comm. Pure Appl. Math. 42 (1989), 687–701; Rev. Mat. Iberoamericana 2 (1986), 235–261; J. Statis. Phys. 46 (1987), 1115–1142.

    Google Scholar 

  80. McKean, H.: Lecture Notes in Math. 755, Springer, New York 1980, pp. 83–200.

    Google Scholar 

  81. Mulase, M.: J. Diff. Geom. 19 (1984), 403–430; Adv. in Math. 54 (1984), 57–66.

    Google Scholar 

  82. Mulase, M.: Invent. Math. 92 (1988), 1–46.

    Google Scholar 

  83. Mulase, M.: Algebraic Analysis, Vol. 2, Academic Press, N.Y., 1988, pp. 473–492.

    Google Scholar 

  84. Mumford, D.: Lectures on Theta, I and II, Birkhauser, Boston, 1983 and 1984.

    Google Scholar 

  85. Newell, A.: Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985.

    Google Scholar 

  86. Newell, A., Tatiu, T., Tabor, M., and Zeng Yanbo, Soliton Mathematics, Univ. Montreal Press, 1986.

  87. Newton, R.: Conf. Inverse Scattering, SIAM, Philadelphia, 1983, pp. 1–74.

  88. Nijhoff, F.: Physica 31D (1988), 339–388.

    Google Scholar 

  89. Nijhoff, F., Quispel, G., Van der Linden, J., and Capel, H.: Physica 119A (1983), 101–142.

    Google Scholar 

  90. Nijhoff, F., Van der Linden, J., Quispel, G., Capel, H., and Velthuizen, J.: Physica 116A (1982), 1–33.

    Google Scholar 

  91. Nijhoff, F.: Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987, pp. 150–181.

    Google Scholar 

  92. Nijhoff, F., Capel, H., Wiersma, G., and Quispel, G.: Phys. Lett. 103A (1984), 293–297.

    Google Scholar 

  93. Nijhoff, F., Capel, H., and Wiersma, G., Lecture Notes in Phys. 239, Springer, New York, 1985, pp. 263–302.

    Google Scholar 

  94. Novikov, S., Manakov, S., Pitaevskij, L., and Zakharov, V.: Theory of Solitons, Plenum, New York, 1984.

    Google Scholar 

  95. Novikov, S. and Khenkin, G.: Uspekhi Mat. Nauk, 42 (1987), 93–152.

    Google Scholar 

  96. Ohta, Y., Satsuma, J., Takahashi, D., and Tokihiro, T.: Prog. Theoret. Phys. Supp. 94 (1988), 210–241.

    Google Scholar 

  97. Oishi, S.: J. Phys. Soc. Japan, 48 (1980), 639–646; 47 (1979), 1341–1346 and 319–325; Mem. School Sci. Engin., Waseda Univ. 46 (1982), 191–225.

    Google Scholar 

  98. Poppe, C.: Physica 13D (1984), 137–160.

    Google Scholar 

  99. Poppe, C.: Inv. Probs. 5 (1989), 613–630.

    Google Scholar 

  100. Poppe, C. and Sattinger, D.: Publ. RIMS, Kyoto Univ. 24 (1988), 505–538.

    Google Scholar 

  101. Post, G., Proefschrift, Univ. Twente, 1988.

  102. Pressley, A. and Segal, G.: Loop Groups, Oxford Univ. Press, 1986.

  103. Previato, E.: Duke Math. J. 52 (1985), 329–377.

    Google Scholar 

  104. Previato, E.: Contemp. Math., Amer. Math. Soc. 64 (1987), pp. 153–180.

    Google Scholar 

  105. Previato, E. and Wilson, G.: Proc. Symp. Pure Math. 49, Amer. Math. Soc., 1989, pp. 553–569.

    Google Scholar 

  106. Ratiu, T.: Lecture Notes in Math. 775, Springer, New York, 1980, pp. 219–257.

    Google Scholar 

  107. Reyman, A.: J. Soviet Math. 19 (1982), 1507–1545.

    Google Scholar 

  108. Reyman, A. and Semenov Tian Shansky, M.: Invent. Math. 54 (1979), 81–100; 63 (1981), 423–432.

    Google Scholar 

  109. Reyman, A. and Semenov Tian Shansky, M.: J. Soviet Math. 31 (1985), 3399–3410.

    Google Scholar 

  110. Reyman, A. and Semenov Tian Shansky, M.: J. Soviet Math. Dokl. 21 (1980), 630–634.

    Google Scholar 

  111. Reyman, A.: J. Soviet Math. 30 (1985), 2319–2325.

    Google Scholar 

  112. Sabatier, P.: Inv. Problems, Academic Press, New York, 1987, pp. 1–18.

    Google Scholar 

  113. Santini, P., Ablowitz, M., and Fokas, A.: J. Math. Phys. 25 (1984), 2614–2619.

    Google Scholar 

  114. Sattinger, D.: Stud. Appl. Math. 72 (1985), 65–86.

    Google Scholar 

  115. Sattinger, D.: Contemp. Math., Amer. Math. Soc. 56 (1986), pp. 369–433.

    Google Scholar 

  116. Sattinger, D., Proc. Inter. Conf. Diff. Eqs., Ohio Univ. Press, 1989, pp. 361–373.

  117. Sattinger, D.: Flat connections and integrable systems, to appear.

  118. Sattinger, D. and Zurkowski, V.: Dynamics of Infinite Dimensional Systems, Springer, New York, 1987. pp. 273–300 and Physica 26D (1987), 225–250.

    Google Scholar 

  119. Sattinger, D.: Flat connections and scattering theory on the line, to appear.

  120. Sattinger, D.: Diff. Integral Equations 1 (1988), 157–172.

    Google Scholar 

  121. Segal, G. and Wilson, G.: Publ. IHES, 61 (1984), 5–65.

    Google Scholar 

  122. Semenov Tian Shansky, M.: Publ. RIMS, Kyoto Univ. 21 (1985), 1237–1260.

    Google Scholar 

  123. Shiota, T.: Invent. Math. 83 (1986), 333–382.

    Google Scholar 

  124. Suzuki, T.: J. Fac. Sci., Univ. Tokyo, 32 (1985), 223–271; Inverse Probs. 4 (1988), 273–299.

    Google Scholar 

  125. Symes, W.: Invent. Math. 59 (1980), 13–51; Physica 1D (1980), 339–374.

    Google Scholar 

  126. Takasaki, K.: Proc. Symp. Pure Math. 49, Amer. Math. Soc., 1989, pp. 143–168.

    Google Scholar 

  127. Ten Kroode, A.: CWI Syllabus, 1984.

  128. Ueno, K.: Infinite Dimensional Groups with Applications, Springer, New York, 1985, pp. 335–353.

    Google Scholar 

  129. Ueno, K.: Vertex Operators in Mathematics and Physics, Springer, New York, 1985, pp. 291–302.

    Google Scholar 

  130. Wadati, M. and Sawada, K.: J. Phys. Soc. Japan 48 (1980), 312–318 and 319–325.

    Google Scholar 

  131. Wickerhauser, M.: Comm. Math. Phys. 108 (1987), 67–89; J. Math. Phys. 29 (1988), 2300–2302.

    Google Scholar 

  132. Wilson, G.: Math. Proc. Cambridge Phil. Soc. 86 (1979), 131–143; Phil. Trans. Royal Soc. London, A315 (1985), 393–404; Ergodic Theory Dynamical Systems 1 (1981), 361–380; Quart. J. Math. Oxford, Ser. 2 32 (1981), 491–512.

    Google Scholar 

  133. Wilson, G.: CR Acad. Sci. Paris, 299 (1984), 587–590.

    Google Scholar 

  134. Witten, E.: Comm. Math. Phys. 113 (1988), 529–600.

    Google Scholar 

  135. Zachary, W.: J. Math. Anal. Appl. 117 (1986), 449–495.

    Google Scholar 

  136. Zakharov, V. and Shabat, A.: Funct. Anal. Appl. 8 (1974), 226–234; 13 (1979), 166–174.

    Google Scholar 

  137. Zakharov, V.: Lecture Notes in Phys. 153, Springer, New York, 1982, pp. 190–216.

    Google Scholar 

  138. Zakharov, V.: Solitons, Topics Current Physics 17, Springer, New York, 1980, pp. 243–285.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 61801, Urbana, IL, U.S.A.

    Robert Carroll

Authors
  1. Robert Carroll
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carroll, R. On the ubiquitous Gelfand-Levitan-Marčenko (GLM) equation. Acta Appl Math 18, 99–144 (1990). https://doi.org/10.1007/BF00046563

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00046563

AMS subject classifications (1980)

Key words

Search

Navigation