Elliptic solitons, Fuchsian equations, and algorithms

Brezhnev, Yu.

doi:10.1090/S1061-0022-2013-01253-2

Elliptic solitons, Fuchsian equations, and algorithms
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by Yu. V. Brezhnev
Translated by: A. Plotkin

St. Petersburg Math. J. 24 (2013), 555-574

DOI: https://doi.org/10.1090/S1061-0022-2013-01253-2

Published electronically: May 24, 2013

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Abstract:

It is shown how the elliptic finite-gap potentials of the Schrödinger equation give rise to a family of solvable linear differential equations of the Fuchs class on the plane and on the torus: the latter case cannot be integrated via realizations of the Zinger–Kovacic type algorithms known in the Picard–Vessiot theory. For the arising Fuchsian equations, monodromy groups and their representations are constructed, the differential Galois group is described, together with a (recursive) method for calculation of the objects involved therein.

References

N. I. Ahiezer, Èlementy teorii èllipticheskikh funktsiĭ, Izdat. “Nauka”, Moscow, 1970 (Russian). Second revised edition]. MR 0288319
L. M. Berkovich, Factorization and transformations of differential equations, R&C Dynamics, Moscow, 2002. (Russian)
A. P. Veselov and A. B. Shabat, A dressing chain and the spectral theory of the Schrödinger operator, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 1–21, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 2, 81–96. MR 1251164, DOI 10.1007/BF01085979
I. M. Gel′fand and L. A. Dikiĭ, Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations, Uspehi Mat. Nauk 30 (1975), no. 5(185), 67–100 (Russian). MR 0508337
B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
A. R. Its and V. B. Matveev, Schrödinger operators with the finite-band spectrum and the $N$-soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz. 23 (1975), no. 1, 51–68 (Russian, with English summary). MR 479120
E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen. 6. Aufl.; Teil II: Partielle Differentialgleichungen erster Ordnung für eine gesuchte Funktion. 4. Aufl, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Band 18, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1959 (German). MR 0106302
Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, New York-Heidelberg, 1972. MR 0349820
H. Poincaré, Sur les groupes des équations linéaires, Acta Math. 4 (1884), no. 1, 201–312 (French). MR 1554639, DOI 10.1007/BF02418420
Yu. N. Sirota and A. O. Smirnov, The Heun equation and the Darboux transformation, Mat. Zametki 79 (2006), no. 2, 267–277 (Russian, with Russian summary); English transl., Math. Notes 79 (2006), no. 1-2, 244–253. MR 2249115, DOI 10.1007/s11006-006-0027-5
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
N. V. Ustinov and Yu. V. Brezhnev, On the $\Psi$-function for finite-gap potentials, Uspekhi Mat. Nauk 57 (2002), no. 1(342), 167–168 (Russian); English transl., Russian Math. Surveys 57 (2002), no. 1, 165–167. MR 1914555, DOI 10.1070/RM2002v057n01ABEH000488
P. B. Acosta-Humánez, Galoisian approach to supersymmetric quantum mechanics. The integrability analysis of the Schrödinger equation by means of differential Galois theory, Verlag Dr. Muller, 2010.
Primitivo B. Acosta-Humánez, Juan J. Morales-Ruiz, and Jacques-Arthur Weil, Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys. 67 (2011), no. 3, 305–374. MR 2846216, DOI 10.1016/S0034-4877(11)60019-0
Acta Applicandae Mathematicae 36 (1994), 1–308 (all vol.).
E. D. Belokolos, A. I. Bobenko, V. Z. Enol′skii, A. R. Its, and V. B. Matveev, Algebro-geometric approach to nonlinear integrable equations, Springer-Verlag, 1994.
Frits Beukers and Alexa van der Waall, Lamé equations with algebraic solutions, J. Differential Equations 197 (2004), no. 1, 1–25. MR 2030146, DOI 10.1016/j.jde.2003.10.017
F. Beukers, Unitary monodromy of Lamé differential operators, Regul. Chaotic Dyn. 12 (2007), no. 6, 630–641. MR 2373163, DOI 10.1134/S1560354707060068
Yurii V. Brezhnev, What does integrability of finite-gap or soliton potentials mean?, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1867, 923–945. MR 2377680, DOI 10.1098/rsta.2007.2056
Yurii V. Brezhnev, Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials, Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics, Contemp. Math., vol. 563, Amer. Math. Soc., Providence, RI, 2012, pp. 1–31. MR 2905627, DOI 10.1090/conm/563/11162
G. Darboux, Sur une équation linéaire, Compt. Rend. Acad. Sci. 94 (1882), no. 25, 1645–1648.
L. A. Dickey, Soliton equations and Hamiltonian systems, 2nd ed., Advanced Series in Mathematical Physics, vol. 26, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1964513, DOI 10.1142/5108
J. Drach, Détermination des cas de réduction de l’équation différentielle $\frac {d^2y}{dx^2}=[\varphi (x)+h] y$, Compt. Rend. Acad. Sci. 168 (1919), 47–50.
Andrew Russell Forsyth, Theory of differential equations. 1. Exact equations and Pfaff’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations, Dover Publications, Inc., New York, 1959. Six volumes bound as three. MR 0123757
Fritz Gesztesy and Rudi Weikard, Picard potentials and Hill’s equation on a torus, Acta Math. 176 (1996), no. 1, 73–107. MR 1395670, DOI 10.1007/BF02547336
Fritz Gesztesy and Rudi Weikard, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math. 181 (1998), no. 1, 63–108. MR 1654775, DOI 10.1007/BF02392748
Pavel Ivanov, On Lamé’s equation of a particular kind, J. Phys. A 34 (2001), no. 39, 8145–8150. MR 1871855, DOI 10.1088/0305-4470/34/39/313
F. Klein, Ueber lineare Differentialgleichungen der zweiten Ordnung, Vorlesung, gehalten im Sommersemester 1894, Göttingen, 1894 (manuscript).
E. R. Kolchin, Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. (2) 49 (1948), 1–42. MR 24884, DOI 10.2307/1969111
Jerald J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), no. 1, 3–43. MR 839134, DOI 10.1016/S0747-7171(86)80010-4
Robert S. Maier, Algebraic solutions of the Lamé equation, revisited, J. Differential Equations 198 (2004), no. 1, 16–34. MR 2037748, DOI 10.1016/j.jde.2003.06.006
Robert S. Maier, Lamé polynomials, hyperelliptic reductions and Lamé band structure, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1867, 1115–1153. MR 2377687, DOI 10.1098/rsta.2007.2063
Juan J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, vol. 179, Birkhäuser Verlag, Basel, 1999. MR 1713573, DOI 10.1007/978-3-0348-8718-2
Marius van der Put and Michael F. Singer, Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. MR 1960772, DOI 10.1007/978-3-642-55750-7
Michael F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations, Amer. J. Math. 103 (1981), no. 4, 661–682. MR 623132, DOI 10.2307/2374045
Michael F. Singer and Felix Ulmer, Liouvillian and algebraic solutions of second and third order linear differential equations, J. Symbolic Comput. 16 (1993), no. 1, 37–73. MR 1237349, DOI 10.1006/jsco.1993.1033
Alexander O. Smirnov, Elliptic solitons and Heun’s equation, The Kowalevski property (Leeds, 2000) CRM Proc. Lecture Notes, vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 287–305. MR 1916788, DOI 10.1090/crmp/032/16
Alexander O. Smirnov, Finite-gap solutions of the Fuchsian equations, Lett. Math. Phys. 76 (2006), no. 2-3, 297–316. MR 2238723, DOI 10.1007/s11005-006-0070-x
Kouichi Takemura, The Heun equation and the Calogero-Moser-Sutherland system. III. The finite-gap property and the monodromy, J. Nonlinear Math. Phys. 11 (2004), no. 1, 21–46. MR 2031210, DOI 10.2991/jnmp.2004.11.1.4
L. W. Thomé, Ueber lineare Differentialgleichungen mit mehrwerthigen algebraischen Coefficienten, J. Reine Angew. Math. 115 (1895), 33–52, 119–149; 119 (1898), 131–147.