[1] Adler, M., van Moerbeke, P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. in Math. 38 (1980),
318-379.
[2] Adler, M., van Moerbeke, P., The complex geometry of the Kowalewski-Painlevé analysis. Invent. Math. 7 (1989), 3-51.
[3] Adler, M., van Moerbeke, P. and Vanhaecke, P., Algebraic integrability, Painlevé geometry and Lie algebras. A series of modern surveys
in mathematics, Volume 47, Springer-Verlag, 2004.
[4] Airault. H., Mc Kean, H.P. and Moser, J., Rational and elliptic solutions of the KdV equation and a related many-body problem. Comm.
Pure Appl. Math. 30 (1977), 94-148.
[5] Arnold, V.I., Mathematical methods in classical mechanics. Springer-Verlag, Berlin-Heidelberg- New York, 1978.
[6] Baker, S., Enolskii, V.Z. and Fordy, A.P., Integrable quartic potentials and coupled KdV equations. Phys. Lett. 201A (1995), 167-174.
[7] Barth, W., Abelian surfaces with (1; 2)polarization. Conf. on alg. geom., Sendai, 1985, Advanced studies in pure mathematics, 10 (1987),
41-84.
[8] Barth, W., Affine parts of abelian surfaces as complete intersections of four quadrics. Math. Ann. 278 (1987), 117-131.
[9] Belokolos, A.I. and Enol’skii, V.Z., Isospectral deformations of elliptic potentials. Russ. Math. Surveys, 44 (1989), 155-156.
[10] Belokolos, A.I., Bobenko, V.Z., Enol’skii, V.Z., Its, A.R. and Matveev, V.B., Algebro-Geometric approach to nonlinear integrable equations.
Springer-Verlag 1994.
[11] Bountis, T., Segur, H. and Vivaldi, F., Integrable hamiltonian systems and the Painlevé property. Phys. Rev. A25 (1982), 1257-1264.
[12] Christiansen, P.L., Eilbeck, J.C., Enolskii, V.Z. and Kostov, N.A., Quasi-periodic solutions of the coupled nonlinear Schrödinger equations.
Proc. R. Soc. Lond. A451 (1995), 685-700.
[13] Conte, R., Musette, M. and Verhoeven, C., Completeness of the cubic and quartic Hénon-Heiles hamiltonians. Theor. Math. Phys. 144
(2005), 888-898.
[14] Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B. and Leykin, D.V., Linear r-matrix algebra for systems separable in parabolic coordinates. Phys.
Lett. 180A (1993), 208-214.
[15] Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B. and Tsiganov, A.V., Linear r-matrix algebra for classical separable systems. J. Phys. A.: Math.
Gen. 27 (1994), 567-578.
[16] Eilbeck, J.C., Enolskii, V.Z., Elliptic solutions and blow-up in an integrable Hénon-Heiles system. Proc. Roy. Soc. Edinburgh, A124 (1994),
1151-1164.
[17] Grammaticos, B. Dorozzi, B., Ramani, A., Integrability of hamiltonians with third and fourth-degree polynomial potentials. J. Math. Phys.
24 (1983), 2289-2295.
[18] Griffiths, P.A., Harris,J., Principles of algebraic geometry.Wiley-Interscience 1978.
[19] Haine, L., Geodesic flow on SO(4) and Abelian surfaces. Math. Ann. 263 (1983), 435-472.
[20] Halphen, G. H., Mémoire sur la réduction des équations différentielles linéaires aux formes intégrales. Mémoires présentés par divers
savants à l’Académie des sciences de l’Institut de France 28, 1-307, 1884.
[21] Hermite, C., Oeuvres de Charles Hermites. Vol. III, Paris, Gauthier-Villar, 1912.
[22] Hietarinta, J., Classical versus quantum integrability. J. Math. Phys. 25 (1984), 1833-1840.
[23] Hietarinta, J., Direct methods for the search of the second invariant. Phys. Rep. 147 (1987), 87-154.
[24] Kasperczuk, S., Integrability of the Yang-Mills hamiltonian system. Celes. Mech. and Dyn. Astr. 58 (1994), 387-391. Erratum Celes. Mech. and
Dyn. Astr. 60 (1994), 289.
[25] Kostov, N.A., Quasi-periodical solutions of the integrable dynamical systems related to Hill’s equation. Lett. Math. Phys. 17 (1989), 95-104.
[26] Kowalewski, S., Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1889), 177-232.
[27] Lesfari, A., Abelian surfaces and Kowalewski’s top. Ann. Scient. École Norm. Sup. Paris, 4e série, t.21 (1988), 193-223.
[28] Lesfari, A., Prym varieties and applications. J. Geom. Phys., 58, 1063-1079 (2008).
[29] Lesfari, A., Integrable systems and complex geometry. Lobachevskii Journal of Mathematics, Vol.30, No.4, 292-326 (2009).
[31] Lesfari, A., Introduction à la géométrie algébrique complexe. Hermann, Paris 2015.
[32] McKean, H.P. and van Moerbeke, P., The Spectrum of Hill’s Equation. Invent. Math. 30 (1975), 217-274.
[33] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16 (1975), 197-219.
[34] Moser, J., Geometry of quadrics and spectral theory. Lecture delivred at the symposium in honor of S.S. Chern, Berkeley, 1979. Springer,
Berlin, Heidelberg, New-York 1980.
[35] Mumford, D., On the equations defining abelian varieties I, II, III. Invent. Math. 1 (1966), 287-354, 3 (1967), 75-135, 3 (1967), 215-244.
[36] Mumford, D., Tata lectures on theta I, II. Progress in mathematics. Birkhaüser, Boston, 1983.
[37] Novikov, S.P., The periodic problem for Korteweg-de Vries equation. Funct. Anal. Pril. 8 (1974), 53-66.
[38] Perelomov, A.M., Integrable systems of classical mechanics and Lie algebras. Birkhäuser Verlag 1990.
[39] Ravoson, V., Ramani, A. and Grammaticos, B., Generalized separability for a hamiltonian with nonseparable quartic potential. Phys. Lett.
191A (1994), 91-95.
[40] Smirnov, A. O., Finite-gap elliptic solutions of the KdV equation. Acta Appl. Math. 36 (1994), 125-199.
[41] Tondo, G., On the integrability of stationary and restricted flows of the KdV hierarchy. J. Phys. A: Mat. Gen. 28 (1995), 5097-5115.
[42] Vanhaecke, P., Linearising two-dimensional integrable systems and the construction of action-angle variables. Math. Z. 211 (1992), 265-313.
[43] Vanhaecke, P., Integrable systems in the realm of algebraic geometry. Lecture Notes in Math. 1638, Springer-Verlag, 2001.
[44] van Moerbeke, P. and Mumford, D., The spectrum of difference operators and algebraic curves. Acta Math. 143 (1979), 93-154.
[45] Verhoeven, C., Musette, M. and Conte, R., General solution of hamiltonians withs extend cubic and quartic potentials. Theor. Math. Phys.
134 (2003), 128-138.
[46] Wojciechowski, S., On a Lax-type representation and separability of the anisotropic harmonic oscillator in a radial quartic potential. Lett.
Nuovo Cimento, 41 (1984), 361-369.
[47] Wojciechowski, S., Integrability of one particle in a perturbed central quartic potential. Physica Scripta, 31 (1985), 433-438.
[48] Yoshida, H., Existence of exponentially unstable solutions and the non-integrability of homogeneous hamiltonian. Physica D, 21 (1986),
163-170.
[50] Ziglin, S.L., Branching of solutions and non-existence of first integrals in hamiltonian mechanics II. Funct. Anal. and its appl. 17 (1983),
6-17.
[1] Adler, M., van Moerbeke, P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. in Math. 38 (1980),
318-379.
[2] Adler, M., van Moerbeke, P., The complex geometry of the Kowalewski-Painlevé analysis. Invent. Math. 7 (1989), 3-51.
[3] Adler, M., van Moerbeke, P. and Vanhaecke, P., Algebraic integrability, Painlevé geometry and Lie algebras. A series of modern surveys
in mathematics, Volume 47, Springer-Verlag, 2004.
[4] Airault. H., Mc Kean, H.P. and Moser, J., Rational and elliptic solutions of the KdV equation and a related many-body problem. Comm.
Pure Appl. Math. 30 (1977), 94-148.
[5] Arnold, V.I., Mathematical methods in classical mechanics. Springer-Verlag, Berlin-Heidelberg- New York, 1978.
[6] Baker, S., Enolskii, V.Z. and Fordy, A.P., Integrable quartic potentials and coupled KdV equations. Phys. Lett. 201A (1995), 167-174.
[7] Barth, W., Abelian surfaces with (1; 2)polarization. Conf. on alg. geom., Sendai, 1985, Advanced studies in pure mathematics, 10 (1987),
41-84.
[8] Barth, W., Affine parts of abelian surfaces as complete intersections of four quadrics. Math. Ann. 278 (1987), 117-131.
[9] Belokolos, A.I. and Enol’skii, V.Z., Isospectral deformations of elliptic potentials. Russ. Math. Surveys, 44 (1989), 155-156.
[10] Belokolos, A.I., Bobenko, V.Z., Enol’skii, V.Z., Its, A.R. and Matveev, V.B., Algebro-Geometric approach to nonlinear integrable equations.
Springer-Verlag 1994.
[11] Bountis, T., Segur, H. and Vivaldi, F., Integrable hamiltonian systems and the Painlevé property. Phys. Rev. A25 (1982), 1257-1264.
[12] Christiansen, P.L., Eilbeck, J.C., Enolskii, V.Z. and Kostov, N.A., Quasi-periodic solutions of the coupled nonlinear Schrödinger equations.
Proc. R. Soc. Lond. A451 (1995), 685-700.
[13] Conte, R., Musette, M. and Verhoeven, C., Completeness of the cubic and quartic Hénon-Heiles hamiltonians. Theor. Math. Phys. 144
(2005), 888-898.
[14] Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B. and Leykin, D.V., Linear r-matrix algebra for systems separable in parabolic coordinates. Phys.
Lett. 180A (1993), 208-214.
[15] Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B. and Tsiganov, A.V., Linear r-matrix algebra for classical separable systems. J. Phys. A.: Math.
Gen. 27 (1994), 567-578.
[16] Eilbeck, J.C., Enolskii, V.Z., Elliptic solutions and blow-up in an integrable Hénon-Heiles system. Proc. Roy. Soc. Edinburgh, A124 (1994),
1151-1164.
[17] Grammaticos, B. Dorozzi, B., Ramani, A., Integrability of hamiltonians with third and fourth-degree polynomial potentials. J. Math. Phys.
24 (1983), 2289-2295.
[18] Griffiths, P.A., Harris,J., Principles of algebraic geometry.Wiley-Interscience 1978.
[19] Haine, L., Geodesic flow on SO(4) and Abelian surfaces. Math. Ann. 263 (1983), 435-472.
[20] Halphen, G. H., Mémoire sur la réduction des équations différentielles linéaires aux formes intégrales. Mémoires présentés par divers
savants à l’Académie des sciences de l’Institut de France 28, 1-307, 1884.
[21] Hermite, C., Oeuvres de Charles Hermites. Vol. III, Paris, Gauthier-Villar, 1912.
[22] Hietarinta, J., Classical versus quantum integrability. J. Math. Phys. 25 (1984), 1833-1840.
[23] Hietarinta, J., Direct methods for the search of the second invariant. Phys. Rep. 147 (1987), 87-154.
[24] Kasperczuk, S., Integrability of the Yang-Mills hamiltonian system. Celes. Mech. and Dyn. Astr. 58 (1994), 387-391. Erratum Celes. Mech. and
Dyn. Astr. 60 (1994), 289.
[25] Kostov, N.A., Quasi-periodical solutions of the integrable dynamical systems related to Hill’s equation. Lett. Math. Phys. 17 (1989), 95-104.
[26] Kowalewski, S., Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1889), 177-232.
[27] Lesfari, A., Abelian surfaces and Kowalewski’s top. Ann. Scient. École Norm. Sup. Paris, 4e série, t.21 (1988), 193-223.
[28] Lesfari, A., Prym varieties and applications. J. Geom. Phys., 58, 1063-1079 (2008).
[29] Lesfari, A., Integrable systems and complex geometry. Lobachevskii Journal of Mathematics, Vol.30, No.4, 292-326 (2009).
[31] Lesfari, A., Introduction à la géométrie algébrique complexe. Hermann, Paris 2015.
[32] McKean, H.P. and van Moerbeke, P., The Spectrum of Hill’s Equation. Invent. Math. 30 (1975), 217-274.
[33] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16 (1975), 197-219.
[34] Moser, J., Geometry of quadrics and spectral theory. Lecture delivred at the symposium in honor of S.S. Chern, Berkeley, 1979. Springer,
Berlin, Heidelberg, New-York 1980.
[35] Mumford, D., On the equations defining abelian varieties I, II, III. Invent. Math. 1 (1966), 287-354, 3 (1967), 75-135, 3 (1967), 215-244.
[36] Mumford, D., Tata lectures on theta I, II. Progress in mathematics. Birkhaüser, Boston, 1983.
[37] Novikov, S.P., The periodic problem for Korteweg-de Vries equation. Funct. Anal. Pril. 8 (1974), 53-66.
[38] Perelomov, A.M., Integrable systems of classical mechanics and Lie algebras. Birkhäuser Verlag 1990.
[39] Ravoson, V., Ramani, A. and Grammaticos, B., Generalized separability for a hamiltonian with nonseparable quartic potential. Phys. Lett.
191A (1994), 91-95.
[40] Smirnov, A. O., Finite-gap elliptic solutions of the KdV equation. Acta Appl. Math. 36 (1994), 125-199.
[41] Tondo, G., On the integrability of stationary and restricted flows of the KdV hierarchy. J. Phys. A: Mat. Gen. 28 (1995), 5097-5115.
[42] Vanhaecke, P., Linearising two-dimensional integrable systems and the construction of action-angle variables. Math. Z. 211 (1992), 265-313.
[43] Vanhaecke, P., Integrable systems in the realm of algebraic geometry. Lecture Notes in Math. 1638, Springer-Verlag, 2001.
[44] van Moerbeke, P. and Mumford, D., The spectrum of difference operators and algebraic curves. Acta Math. 143 (1979), 93-154.
[45] Verhoeven, C., Musette, M. and Conte, R., General solution of hamiltonians withs extend cubic and quartic potentials. Theor. Math. Phys.
134 (2003), 128-138.
[46] Wojciechowski, S., On a Lax-type representation and separability of the anisotropic harmonic oscillator in a radial quartic potential. Lett.
Nuovo Cimento, 41 (1984), 361-369.
[47] Wojciechowski, S., Integrability of one particle in a perturbed central quartic potential. Physica Scripta, 31 (1985), 433-438.
[48] Yoshida, H., Existence of exponentially unstable solutions and the non-integrability of homogeneous hamiltonian. Physica D, 21 (1986),
163-170.
[50] Ziglin, S.L., Branching of solutions and non-existence of first integrals in hamiltonian mechanics II. Funct. Anal. and its appl. 17 (1983),
6-17.
Lesfari, A. (2018). Geometric Study of a Family of Integrable Systems. International Electronic Journal of Geometry, 11(1), 78-92. https://doi.org/10.36890/iejg.545100
AMA
Lesfari A. Geometric Study of a Family of Integrable Systems. Int. Electron. J. Geom. Nisan 2018;11(1):78-92. doi:10.36890/iejg.545100
Chicago
Lesfari, Ahmed. “Geometric Study of a Family of Integrable Systems”. International Electronic Journal of Geometry 11, sy. 1 (Nisan 2018): 78-92. https://doi.org/10.36890/iejg.545100.
EndNote
Lesfari A (01 Nisan 2018) Geometric Study of a Family of Integrable Systems. International Electronic Journal of Geometry 11 1 78–92.
IEEE
A. Lesfari, “Geometric Study of a Family of Integrable Systems”, Int. Electron. J. Geom., c. 11, sy. 1, ss. 78–92, 2018, doi: 10.36890/iejg.545100.
ISNAD
Lesfari, Ahmed. “Geometric Study of a Family of Integrable Systems”. International Electronic Journal of Geometry 11/1 (Nisan 2018), 78-92. https://doi.org/10.36890/iejg.545100.
JAMA
Lesfari A. Geometric Study of a Family of Integrable Systems. Int. Electron. J. Geom. 2018;11:78–92.
MLA
Lesfari, Ahmed. “Geometric Study of a Family of Integrable Systems”. International Electronic Journal of Geometry, c. 11, sy. 1, 2018, ss. 78-92, doi:10.36890/iejg.545100.
Vancouver
Lesfari A. Geometric Study of a Family of Integrable Systems. Int. Electron. J. Geom. 2018;11(1):78-92.