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Picard-Vessiot theory and integrability. (English) Zbl 1372.12006

In this paper the author presents a survey of recent results on the integrability of dynamical systems obtained with the help of differential Galois theory. The survey is a continuation of the previous work [Contemp. Math. 509, 143–220 (2010; Zbl 1294.37024)] of the author and J.-P. Ramis and therefore only new works not included in the previous review are considered in it. These results are described only at the level of ideas without going into details. The paper consists of the following paragraphs:
1. Picard-Vessiot theory
2. Integrability of Hamiltonian systems
3. Some spectral problems
In the first section all the results, methods and ideas from the differential Galois theory necessary for understanding the reader are described. In the second section new results are presented for Hamiltonian systems. They show the connection between the commutativity of the component of the identity of the Galois group of variational equations and the integrability of the Hamiltonian system. The third section is mainly devoted to the description of the algebraic spectrum for various variants of the Schrödinger operator. The cases of the harmonic oscillator, the Liouville potential and the Coulomb potential are considered.
\[ \frac{d^2\psi}{dx^2}=\left( \frac{1}{4}\omega^2 x^2-E \right)\psi ,\quad E_n=\left( \frac{1}{2}+n \right)\omega,\;n\in \mathbb{Z}\quad(\text{harmonic oscillator}). \]
\[ \frac{d^2\psi}{dx^2}=\left(e^{-2x}-E\right)\psi ,\quad E_n=-(\tfrac{1}{2}+n)^2,\;n\in \mathbb{Z}\quad \text{(Liouville potential)}. \]
\[ \frac{d^2\psi}{dr^2}=\left(\frac{l(l+1)}{r^2}-\frac{e^2}{r}+\frac{e^4}{4(l+1)^2}-E\right)\psi,\;l\in \mathbb{Z}\quad E_{ln}=\frac{e^4}{4(l+1)^2}\lambda _{ln}, \]
\[ \lambda_{ln}\in \left\{ 1-\left(\frac{l+1}{l+1+n} \right)^2: n\in\mathbb{Z}_+ \right\}\bigcup \left\{ 1-\left( \frac{l+1}{l-n} \right)^2: n\in\mathbb{Z}_+ \right\}\quad\text{(Coulomb potential)}\,. \]

MSC:

12H05 Differential algebra
32S65 Singularities of holomorphic vector fields and foliations
37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
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[1] Morales-Ruiz, J.; Ramis, J.-P., Integrability of dynamical systems through differential Galois theory: a practical guide, (Acosta-Humánez, P.-B.; Marcellán, F., Differential Algebra, Complex Analysis and Orthogonal Polynomials. Differential Algebra, Complex Analysis and Orthogonal Polynomials, Contemp. Math., vol. 509 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 143-220 · Zbl 1294.37024
[2] Acosta-Humánez, P.; Morales-Ruiz, J.-J.; Weil, J.-A., Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys., 67, 305-374 (2011) · Zbl 1238.81090
[4] Acosta-Humánez, P. B.; Pantazi, C., Darboux integrals for Schrödinger planar vector fields via Darboux transformations, SIGMA Symmetry Integrability Geom. Methods Appl., 8, 043 (2012) · Zbl 1281.34045
[5] Weil, J.-A., Méthodes effectives en théorie de Galois différentielle et applications à l’intégrabilité de systèmes dynamiques (2013), Université de Limoges, (Habilitation thesis)
[6] Humphreys, J. E., Linear Algebraic Groups (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0471.20029
[7] Borel, A., Linear Algebraic Groups (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0726.20030
[8] Picard, E., Sur les groupes de transformation des équations différentielles linéaires, C. R. Acad. Sci., Paris, 96, 1131-1134 (1883) · JFM 15.0258.02
[9] Picard, E., Sur équations différentielles et les groupes algébriques des transformation, Ann. Fac. Sci. Univ. Toulouse (1), 1, A1-A15 (1887) · JFM 19.0308.01
[10] Picard, E., Traité d’Analyse, Tome III (1928), Gauthiers-Villars: Gauthiers-Villars Paris · JFM 54.0450.09
[11] Vessiot, M. E., Sur l’intégration des équations différentielles linéaires, Ann. Sci. Éc. Norm. Supér. (3), 9, 197-280 (1892) · JFM 24.0283.01
[12] van der Put, M.; Singer, M., Galois Theory of Linear Differential Equations (2003), Springer: Springer Berlin · Zbl 1036.12008
[13] Kolchin, E., Differential Algebra and Algebraic Groups (1973), Academic Press · Zbl 0264.12102
[14] Kaplansky, I., An Introduction to Differential Algebra (1957), Hermann · Zbl 0083.03301
[15] Singer, M., (MacCallum, M. A.H.; Mikhalov, A. V., Introduction to the Galois Theory of Linear Differential Equations Algebraic Theory of Differential Equations. Introduction to the Galois Theory of Linear Differential Equations Algebraic Theory of Differential Equations, London Mathematical Society Lecture Note Series, vol. 357 (2009), Cambridge University Press), 1-82 · Zbl 1176.12005
[16] Martinet, J.; Ramis, J. P., Théorie de Galois differentielle et resommation, (Tournier, E., Computer Algebra and Differential Equations (1989), Academic Press: Academic Press London), 117-214
[17] Liouville, J., Mémoire sur l’intégration d’une classe d’équations différentielles du second ordre en quantités finies explicites, J. Math. Pures Appl. (1), 4, 423-456 (1839)
[18] Blázquez-Sanz, D.; Morales-Ruiz, J. J., Differential Galois theory of algebraic Lie-Vessiot systems, (Differential Algebra, Complex Analysis and Orthogonal Polynomials. Differential Algebra, Complex Analysis and Orthogonal Polynomials, Contemp. Math., vol. 509 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-58 · Zbl 1200.34109
[19] Blázquez-Sanz, D., Differential Galois Theory and Lie-Vessiot Sytems (2008), VDM Verlag, (Ph.D. thesis)
[20] Morales-Ruiz, J. J.; Ramis, J. P., Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal., 8, 33-96 (2001) · Zbl 1140.37352
[21] Morales-Ruiz, J., (Differential Galois Theory and Non-integrability of Hamiltonian Systems. Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, vol. 179 (1999), Birkhäuser: Birkhäuser Basel) · Zbl 0934.12003
[22] Acosta-Humánez, P.; Blázquez-Sanz, D., Non-integrability of some hamiltonian systems with rational potential, Discrete Contin. Dyn. Syst. Ser. B, 10, 265-293 (2008) · Zbl 1175.37060
[23] Umemura, H., Differential Galois theory of infinite dimension, Nagoya Math. J., 144, 59-135 (1996) · Zbl 0878.12002
[24] Malgrange, B., Le groupoide de Galois d’un feuilletage, Essays on Geometry and Related Topics, Vols. 1, 2. Essays on Geometry and Related Topics, Vols. 1, 2, Monogr. Enseign. Math., 38, 2, 465-501 (2001) · Zbl 1033.32020
[25] Malgrange, B., On the non linear Galois differential theory, Chin. Ann. Math. Ser., B 23, 219-226 (2002) · Zbl 1009.12005
[26] Casale, G., Sur le groupoïde de Galois d’un feuilletage (2004), Univ. Toulouse, (Ph.D. thesis)
[27] Casale, G., Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres, Ann. Inst. Fourier, 56, 735-779 (2006) · Zbl 1155.32020
[28] Casale, G., The Galois groupoid of Picard Painlevé VI equation, Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and Their Deformations, Painlevé Hierarchies. Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and Their Deformations, Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, B2, 15-20 (2007) · Zbl 1219.34113
[29] Casale, G., Le groupe de Galois de \(P^1\) et son irreductibilité, Comment. Math. Helv., 83, 471-519 (2008) · Zbl 1163.34060
[30] Cassidy, P.; Singer, M., Galois theory of parameterized differential equations and linear differential algebraic groups, (Bertrand, D.; Enriquez, B.; Mitschi, C.; Sabbah, C.; Schaefke, R., Differential Equations and Quantum Groups. Differential Equations and Quantum Groups, IRMA Lectures in Mathematics and Theoretical Physics, vol. 9 (2006), EMS Publishing house), 113-157
[31] Morales-Ruiz, J. J.; Ramis, J. P.; Simó, C., Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations, Ann. Sci. Éc. Norm. Supér., 40, 845-884 (2007) · Zbl 1144.37023
[32] Morales-Ruiz, J. J.; Ramis, J. P., A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8, 113-120 (2001) · Zbl 1140.37353
[33] Yoshida, H., A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D, 29, 128-142 (1987) · Zbl 0659.70012
[34] Aparicio-Monforte, A., Méthodes effectives pour l’intégrabilité des systèmes dynamiques (December 2010), Université de Limoges, (Ph.D. Thesis)
[35] Tsygvintsev, A. V., Non-existence of new meromorphic first integrals in the planar three-body problem, Celestial Mech. Dynam. Astronom., 86, 237-247 (2003) · Zbl 1062.70024
[36] Aparicio-Monforte, A.; Weil, J.-A., A reduction method for higher order variational equations of Hamiltonian systems. Symmetries and related topics in differential and difference equations, (Contemp. Math., vol. 549 (2011), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-15 · Zbl 1243.37048
[37] Aparicio-Monforte, A.; Barkatou, M.; Simon, S.; Weil, J.-A., Formal first integrals along solutions of differential systems I. ISSAC 2011, (Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation (2011), ACM: ACM New York), 19-26 · Zbl 1323.34097
[38] Aparicio-Monforte, A.; Weil, J.-A., A reduced form for linear differential systems and its application to integrability of Hamiltonian systems, J. Symbolic Comput., 47, 192-213 (2012) · Zbl 1269.37026
[39] Aparicio-Monforte, A.; Compoint, E.; Weil, J.-A., A characterization of reduced forms of linear differential systems, J. Pure Appl. Algebra, 217, 1504-1516 (2013) · Zbl 1272.12016
[42] Ayoul, M.; Zung, N. T., Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris, 348, 1323-1326 (2010) · Zbl 1210.37076
[43] Combot, T., A note on algebraic potentials and Morales-Ramis theory, Celestial Mech. Dynam. Astronom., 115, 397-404 (2013) · Zbl 1266.70032
[44] Combot, T., Non-integrability of a self-gravitating Riemann liquid ellipsoid, Regul. Chaotic Dyn., 18, 497-507 (2013) · Zbl 1417.37195
[45] Rosemann, S.; Schöbel, K., Open problems in the theory of finite-dimensional integrable systems and related fields, J. Geom. Phys. (2014), submitted for publication
[46] Combot, T., Non-intégrabilité algébrique et méromorphe de problèmes de n corps et de potentiels homogènes de degré −1 (2013), Univ. Paris 7, (Ph.D. thesis)
[47] Combot, T., Integrability conditions at order 2 for homogeneous potentials of degree −1, Nonlinearity, 26, 95-120 (2013) · Zbl 1277.37088
[48] Duval, G.; Maciejewski, A. J., Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Ann. Inst. Fourier, 59, 2839-2890 (2009) · Zbl 1196.37096
[50] Combot, T.; Koutschan, C., Third order integrability conditions for homogeneous potentials of degree −1, J. Math. Phys., 53 (2012), 082704-26 · Zbl 1331.70050
[51] Combot, T., Non-integrability of the equal mass n-body problem with non-zero angular momentum, Celestial Mech. Dynam. Astronom., 114, 319-340 (2012) · Zbl 1266.70018
[52] Casale, G.; Duval, G.; Maciejewski, A. J.; Przybylska, M., Integrability of Hamiltonian systems with homogeneous potentials of degree zero, Phys. Lett. A, 374, 448-452 (2010) · Zbl 1235.70036
[53] Maciejewski, A. J.; Prybylska, M., Integrable deformations of integrable Hamiltonian systems, Phys. Lett. A, 376, 80-93 (2011) · Zbl 1255.70016
[55] Kovacic, J., An algorithm for solving second order linear homogeneus differential equations, J. Symbolic. Comput., 2, 3-43 (1986) · Zbl 0603.68035
[60] Waters, T., Regular and irregular geodesics on spherical harmonic surfaces, Physica D, 241, 543-552 (2012) · Zbl 1250.53039
[61] Waters, T., Non-integrability of geodesic flow on certain algebraic surfaces, Phys. Lett. A, 376, 1442-1445 (2012) · Zbl 1260.53137
[62] Combot, T.; Waters, T., Integrability conditions of geodesic flow on homogeneous Monge manifolds, Ergodic Theory Dynam. Systems (2014), in press
[64] Basu, P.; Pando Zayas, L. A., Analytic non-integrability in string theory, Phys. Rev. D, 48, 046006 (2011)
[65] Stepanchuk, A.; Tseytlin, A. A., On (non) integrability of classical strings in p-brane backgrounts, J. Phys. A: Math. Ther., 46, 125401-125421 (2013) · Zbl 1267.81260
[66] Ghosh, A., Time-dependent systems and chaos in string theory (2012), Univ. Kentucky, (Ph.D. thesis)
[68] Basu, P.; Das, D.; Ghosh, A.; Panda Zayas, L. A., Chaos around holographic Regge trajectories, J. High Energy Phys., 2012, 5, 077 (2012) · Zbl 1348.83022
[71] Basu, P.; Das, D.; Ghosh, A., Integrability lost: chaotic dynamics of classical strings on a confining holographic background, Phys. Lett. B, 699, 388-393 (2011)
[72] Mason, L. J.; Woodhouse, N. M.J., Integrability, Self-Duality and Twistor Theory (1996), Clarendon Press: Clarendon Press Oxford · Zbl 0856.58002
[73] Morales-Ruiz, J. J., A Remark about the Painlevé transcendents, Théories asymptotiques et équations de Painlevé, S.M.F. Théories asymptotiques et équations de Painlevé, S.M.F, Sémin. Congr., 14, 229-235 (2005) · Zbl 1140.37016
[74] Stoyanova, T.; Christov, O., Non-integrability of the second Painlevé equation as a Hamiltonian system, C. R. Acad. Bulg. Sci., 60, 1 (2007) · Zbl 1139.70009
[75] Horozov, E.; Stoyanova, T., Non-integrability of some Painlevé VI-equations and dilogarithms, Regul. Chaotic Dyn., 12, 622-629 (2007) · Zbl 1229.37057
[76] Beisert, N., Review of AdS/CFT integrability: an overview, Lett. Math. Phys., 99, 3 (2012)
[77] Martínez, R.; Simó, C., Non-integrability of Hamiltonian systems through high order variational equations: summary of results and examples, Regul. Chaotic Dyn., 14, 323-348 (2009) · Zbl 1229.37058
[80] Christov, O., Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom, Celestial Mech. Dynam. Astronom., 112, 149-167 (2012) · Zbl 1266.70038
[81] Morales-Ruiz, J. J.; Simó, C.; Simon, S., Algebraic proof of the non-integrability of Hill’s problem, Ergodic Theory Dynam. Systems, 25, 1237-1256 (2005) · Zbl 1078.70014
[82] Landau, L. D.; Lifshitz, E. M., Quantum Mechanics: Non-relativistic Theory (1977), Pergamon Press: Pergamon Press Oxford · Zbl 0178.57901
[83] Darboux, G., Sur une proposition relative aux équations linéaires, C. R. Acad. Sci., 94, 1456-1459 (1882) · JFM 14.0264.01
[84] Darboux, G., Théorie des Surfaces, II (1889), Gauthier-Villars · JFM 21.0744.02
[85] Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0744.35045
[86] Morales-Ruiz, J. J.; Peris, J. M., On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse, VIII, 125-141 (1999) · Zbl 0971.34076
[87] Grotta-Ragazzo, C., Nonintegrability of some Hamiltonian systems, scattering and analytic continuation, Comm. Math. Phys., 166, 255-277 (1994) · Zbl 0814.70009
[88] Rosen, N.; Morse, P. M., On the vibrations of polyatomic molecules, Phys. Rev., 42, 210-217 (1932) · JFM 58.1361.05
[89] Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 121-251 (1979) · Zbl 0388.34005
[90] Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons (1984), Consultants Bureau: Consultants Bureau New York · Zbl 0598.35002
[91] Fauvet, F.; Ramis, J.-P.; Richard-Jung, F.; Thomann, J., Stokes phenomenon for the prolate spheroidal wave equation, Appl. Numer. Math., 60, 1309-1319 (2010) · Zbl 1213.34113
[92] Stachowiak, T.; Przybylska, M., On integrable rational potentials of the Dirac equation, Phys. Lett. A, 377, 833-841 (2013) · Zbl 1298.81083
[93] Maciejewski, A. J.; Przybylska, M.; Stachowiak, T., Full spectrum of the Rabi model. Full spectrum of the Rabi model, Phys. Lett. A, 378, 16-20 (2014) · Zbl 1396.81229
[94] Blázquez-Sanz, D.; Yagasaki, K., Galoisian approach for a Sturm-Liouville problem on the infinite interval, Methods Appl. Anal., 19, 267-288 (2012) · Zbl 1273.34033
[95] Etingof, P.; Rains, E., On algebraically integrable differential operators on an elliptic curve, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 19 pp, Paper 062 · Zbl 1247.37045
[96] Krigorenko, N. V., Algebraic-geometric operators and differential Galois theory, Ukrainian Math. J., 61, 14-29 (2009)
[97] Brezhnev, Yu. V., What does integrability of finite-gap or soliton potentials mean?, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366, 1867, 923-945 (2008) · Zbl 1153.37411
[98] Brezhnev, Yu. V., Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials, (Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics. Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemp. Math., vol. 563 (2012), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-31 · Zbl 1248.81032
[99] Brezhnev, Yu. V., Elliptic solitons, Fuchsian equations, and algorithms, St. Petersburg Math. J., 24, 555-574 (2013) · Zbl 1276.35052
[100] Duval, A.; Loday-Richaud, M., Kovacic’s algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput., 3, 3, 211-246 (1992) · Zbl 0785.12003
[101] Ulmer, F.; Weil, J. A., Note on Kovacic’s algorithm, J. Symbolic Comput., 22, 179-200 (1996) · Zbl 0871.12008
[102] Kimura, T., On Riemann’s equations which are solvable by quadratures, Funkcial. Ekvac., 12, 269-281 (1969) · Zbl 0198.11601
[103] Poole, E. G.C., Introduction to the Theory of Linear Differential Equations (1936), Oxford Univ. Press: Oxford Univ. Press London · JFM 62.1277.01
[104] Chihara, T., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach · Zbl 0389.33008
[105] Whittaker, E. T.; Watson, E. T., A Course of Modern Analysis (1969), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · JFM 45.0433.02
[106] Halphen, G. H., Traité des fonctions elliptiques, Vols. I, II (1888), Gauthier-Villars: Gauthier-Villars Paris
[107] Baldassarri, F., On algebraic solutions of Lamé’s differential equation, J. Differential Equations, 41, 44-58 (1981) · Zbl 0478.34009
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