Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 020, 23 pages      arXiv:1403.1012      https://doi.org/10.3842/SIGMA.2014.020
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

The Sturm-Liouville Hierarchy of Evolution Equations and Limits of Algebro-Geometric Initial Data

Russell Johnson a and Luca Zampogni b
a) Dipartimento di Sistemi e Informatica, Università di Firenze, Italy
b) Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Italy

Received October 17, 2013, in final form February 27, 2014; Published online March 05, 2014

Abstract
The Sturm-Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (2011), 555-591] and includes the Korteweg-de Vries and the Camassa-Holm hierarchies. We discuss some solutions of this hierarchy which are obtained as limits of algebro-geometric solutions. The initial data of our solutions are (generalized) reflectionless Sturm-Liouville potentials [Stoch. Dyn. 8 (2008), 413-449].

Key words: Sturm-Liouville problem; m-functions; zero-curvature equation; hierarchy of evolution equations; recursion system.

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References

  1. Alber M.S., Fedorov Y.N., Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, Inverse Problems 17 (2001), 1017-1042.
  2. Beals R., Sattinger D.H., Szmigielski J., Multipeakons and the classical moment problem, Adv. Math. 154 (2000), 229-257, solv-int/9906001.
  3. Belokolos E.D., Bobenko A.I., Enol'skii V.Z., Its A.R., Matveev V.B., Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994.
  4. Boutet de Monvel A., Egorova I., On solutions of nonlinear Schrödinger equations with Cantor-type spectrum, J. Anal. Math. 72 (1997), 1-20.
  5. Camassa R., Holm D.D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664, patt-sol/9305002.
  6. Craig W., The trace formula for Schrödinger operators on the line, Comm. Math. Phys. 126 (1989), 379-407.
  7. Dubrovin B.A., Matveev V.B., Novikov S.P., Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Russ. Math. Surv. 31 (1976), 59-146.
  8. Egorova I.E., On a class of almost periodic solutions of the KdV equation with a nowhere dense spectrum, Russian Acad. Sci. Dokl. Math. 45 (1993), 290-293.
  9. Egorova I.E., The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, in Spectral operator theory and related topics, Adv. Soviet Math., Vol. 19, Amer. Math. Soc., Providence, RI, 1994, 181-208.
  10. Fabbri R., Johnson R., Zampogni L., Nonautonomous differential systems in two dimensions, in Handbook of Differential Equations: Ordinary Differential Equations, Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 133-268.
  11. Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095-1097.
  12. Gesztesy F., Holden H., Algebro-geometric solutions of the Camassa-Holm hierarchy, Rev. Mat. Iberoam. 19 (2003), 73-142, nlin.SI/0105021.
  13. Gesztesy F., Holden H., Soliton equations and their algebro-geometric solutions. Vol. I. (1+1)-dimensional continuous models, Cambridge Studies in Advanced Mathematics, Vol. 79, Cambridge University Press, Cambridge, 2003.
  14. Gesztesy F., Karwowski W., Zhao Z., Limits of soliton solutions, Duke Math. J. 68 (1992), 101-150.
  15. Johnson R., Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), 54-78.
  16. Johnson R., On the Sato-Segal-Wilson solutions of the K-dV equation, Pacific J. Math. 132 (1988), 343-355.
  17. Johnson R., Moser J., The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), 403-438.
  18. Johnson R., Zampogni L., On the inverse Sturm-Liouville problem, Discrete Contin. Dyn. Syst. 18 (2007), 405-428.
  19. Johnson R., Zampogni L., Description of the algebro-geometric Sturm-Liouville coefficients, J. Differential Equations 244 (2008), 716-740.
  20. Johnson R., Zampogni L., Some remarks concerning reflectionless Sturm-Liouville potentials, Stoch. Dyn. 8 (2008), 413-449.
  21. Johnson R., Zampogni L., On the Camassa-Holm and K-dV hierarchies, J. Dynam. Differential Equations 22 (2010), 331-366.
  22. Johnson R., Zampogni L., Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), 559-586.
  23. Johnson R., Zampogni L., The Sturm-Liouville hierarchy of evolution equations, Adv. Nonlinear Stud. 11 (2011), 555-591.
  24. Johnson R., Zampogni L., The Sturm-Liouville hierarchy of evolution equations. II, Adv. Nonlinear Stud. 12 (2012), 501-532.
  25. Johnson R., Zampogni L., On a Gel'fand-Levitan theory for the Sturm-Liouville operator, Preprint.
  26. Knopp K., Funktionentheorie. II. Anwendungen und Weiterführung der allgemeinen Theorie, de Gruyter, Berlin, 1981.
  27. Kotani S., KdV flow on generalized reflectionless potentials, J. Math. Phys. Anal. Geometry 4 (2008), 490-528.
  28. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), 253-290.
  29. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. II, Comm. Pure Appl. Math. 36 (1983), 571-593.
  30. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. III, Comm. Pure Appl. Math. 36 (1983), 809-829.
  31. Levitan B.M., Approximation of infinite-zone by finite-zone potentials, Math. USSR Izv. 20 (1983), 55-87.
  32. Levitan B.M., On the closure of the set of finite-zone potentials, Math. USSR Sb. 51 (1985), 67-89.
  33. Levitan B.M., Inverse Sturm-Liouville problems, VSP, Zeist, 1987.
  34. Lundina D., Compactness of the set of reflectionless potentials, Teor. Funkts. Funkts. Anal. Prilozh. 44 (1985), 55-66.
  35. Marchenko V.A., Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, Vol. 22, Birkhäuser Verlag, Basel, 1986.
  36. Marchenko V.A., The Cauchy problem for the KdV equation with nondecreasing initial data, in What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 273-318.
  37. Marchenko V.A., Ostrovsky I.V., Approximation of periodic by finite-zone potentials, Sel. Math. Sov. 6 (1987), 101-136.
  38. McKean H.P., The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies, Comm. Pure Appl. Math. 56 (2003), 998-1015.
  39. McKean H.P., van Moerbeke P., The spectrum of Hill's equation, Invent. Math. 30 (1975), 217-274.
  40. Novikov S., Manakov S.V., Pitaevski L.P., Zakharov V.E., Theory of solitons. The inverse scattering method, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1984.
  41. Sato M., Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds, in Random Systems and Dynamical Systems (Kyoto, 1981), RIMS Kokyuroku, Vol. 439, Kyoto, 1981, 30-46.
  42. Segal G., Wilson G., Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. (1985), 5-65.
  43. Venakides S., The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43 (1990), 335-361.
  44. Zampogni L., On algebro-geometric solutions of the Camassa-Holm hierarchy, Adv. Nonlinear Stud. 7 (2007), 345-380.
  45. Zampogni L., On infinite order K-dV hierarchies, J. Appl. Funct. Anal. 4 (2009), 140-170.

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