Abstract
In this paper, we study the Darboux transformation of the Darboux-Treibich-Verdier equation. On the basis of this transformation, we construct a generalization of the Darboux transformation to the case of the Heun equation and to other linear ordinary differential equations of second order. Examples are given.
Similar content being viewed by others
References
V. B. Matveev and V. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, 1991, pp. 1–120.
I. A. Dynnikov and S. P. Novikov, “Discrete spectral symmetries of small-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds,” Uspekhi Mat. Nauk [Russian Math. Surveys], 52 (1997), no. 5, 175–234.
G. Darboux, Lecons sur la theorie generale des surfaces. II, 2 ed., Gauthier-Villars, Paris, 1915.
V. B. Matveev, “Positons: slowly decreasing analogs of solitons,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 131 (2002), no. 1, 44–61.
M. Humi, “Fractional Darboux transformations,” in: E-print math-ph/0202020, 2002.
D. J. Fernandez, B. Mielnik, and J. O. Rosas-Ortiz, and B. F. Samsonov, “Nonlocal SUSE deformations of periodic potentials,” in: E-print quant-ph/0303051, 2003.
J. Weiss, “Period fixed points of Backlund transformations and the KdV equation,” J. Math. Phys., 27 (1986), no. 11, 2647–2656; 28 (1987), no. 9, 2025–2039.
A. P. Veselov and A. B. Shabat, “The dressing chain and the spectral theory of the Schrodinger operator,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 27 (1993), no. 2, 1–21.
G. Darboux, “Sur une equation lineaire,” C. R. Acad. Sci. Paris, 44 (1882), no. 25, 1645–1648.
A. Treibich and J.-L. Verdier, “Solitons elliptiques,” in: Progr. Math. (The Grothendieck Festschrift, vol. 3), vol. 88, Birkhauser, Boston, MA, 1990, pp. 437–480.
A. Treibich and J.-L. Verdier, “Revetements tangentiels et sommes de 4 nombres triangulaires,” C. R. Acad. Sci. Paris. Ser. I, 311 (1990), 51–54.
A. Treibich, “Revetements exceptionnels et sommes de 4 nombres triangulaires,” Duke Math. J., 68 (1992), 217–236.
E. D. Belokolos, A. I. Bobenko, V. B. Matveev, and V. Z. Enol'skii, “Algebro-geometrical principles of superposition of finite-gap solutions of integrable nonlinear equations,” Uspekhi Mat. Nauk [Russian Math. Surveys], 41 (1986), no. 2, 3–42.
E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii (Enol'skii), and A. R. Its, and V. B. Matveev, Algebro-Geometrical Approach to Nonlinear Evolution Equations, Springer Ser. Nonlinear Dynamics, Springer-Verlag, Berlin-Heidelberg-New York, 1994.
E. D. Belokolos and V. Z. Enol'skii, “Verdier elliptic solitons and the Weierstrass reduction theory,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 23 (1989), no. 1, 57–58.
E. D. Belokolos and V. Z. Enol'skii (Enol'skii), “Reduction of the theta function and elliptic finite-gap potentials,” Acta Appl. Math., 36 (1994), 87–117.
A. O. Smirnov, “The elliptic solutions of the Korteweg-de Vries equation,” Mat. Zametki [Math. Notes], 45 (1989), no. 6, 66–73.
A. O. Smirnov, “The elliptic solutions of integrable nonlinear equations,” Mat. Zametki [Math. Notes], 46 (1989), no. 5, 100–102.
A. O. Smirnov, “Finite-gap elliptic solutions of the KdV equation,” Acta Appl. Math., 36 (1994), 125–166.
N. I. Akhiezer, Elements of the Theory of Elliptic Functions [in Russian], Nauka, Moscow, 1970.
V. I. Inozemtsev, “Lax representation with spectral parameter on torus for integrable particle systems,” Lett. Math. Phys., 17 (1989), 11–17.
K. Takemura, “The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method,” Comm. Math. Phys., 235 (2003), 467–494.
K. Takemura, “On the Inozemtsev model,” in: E-print math-ph/0312037, 2003.
A. O. Smirnov, “Elliptic solitons and the Heun equation,” CRM Proc. Lecture Notes, 32 (2002), 287–305.
A. O. Smirnov, “Finite-gap solutions of the Fuchsian equations,” in: E-print math.CA/0310465, 2003.
E. Kamke, E. Kamke, Differentialgleichungen, Losungsmethoden und Losungen, Leipzig, 1959; Russian translation: Fizmatgiz, Moscow, 1961.
H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York-Toronto-London, 1953–55. Russian translation: Fizmatgiz, Moscow, 1967.
Heun's Differential Equations (A. Ronveaux, Ed.), Oxford University Press, Oxford, 1995.
S. Yu. Slavyanov and W. Lay, Special Functions, Oxford University Press, Oxford, 2000.
S. Yu. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on the Analysis of Singularities [in Russian] Nevskii Dialekt, St. Petersburg., 2002.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (M. Abramowitz and I. Stegun, Eds.) National Bureau of Standards, Washington, D.C., 1964. Russian translation: Nauka, Moscow, 1979.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 79, no. 2, 2006, pp. 267–277.
Original Russian Text Copyright © 2006 by Yu. N. Sirota, A. O. Smirnov.
Rights and permissions
About this article
Cite this article
Sirota, Y.N., Smirnov, A.O. The Heun Equation and the Darboux Transformation. Math Notes 79, 244–253 (2006). https://doi.org/10.1007/s11006-006-0027-5
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0027-5