
Previous Article
A note on the Cauchy problem of a modified CamassaHolm equation with cubic nonlinearity
 DCDS Home
 This Issue

Next Article
Topological defects in the abelian Higgs model
Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity
1.  Laboratoire de Mathématiques et d'Informatique (LMI), INSA de Rouen, Avenue de l'Université, 76 801 Saint Etienne du Rouvray Cedex 
2.  Institute of Astronomy, University of Zielona Góra, Licealna 9, PL65417, Zielona Góra, Poland 
References:
[1] 
A. Aparicio Monforte and J.A. Weil, A reduction method for higher order variational equations of Hamiltonian systems, In Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 115. doi: 10.1090/conm/549/10850. 
[2] 
T. Combot, Nonintegrability of the equal mass; nbody problem with nonzero angular momentum, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319340. doi: 10.1007/s105690129417z. 
[3] 
G. Duval and A. J. Maciejewski, Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Annales de l'Institut Fourier, 59 (2009), 28392890. doi: 10.5802/aif.2510. 
[4] 
G. Duval and A. J. Maciejewski, Integrability of Homogeneous potential of degree $k = \pm 2$. An application of higher variational equations, submited, 2012. 
[5] 
J. J. MoralesRuiz and J. P. Ramis, A note on the nonintegrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113120. 
[6] 
E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications Inc., New York, 1960. 
show all references
References:
[1] 
A. Aparicio Monforte and J.A. Weil, A reduction method for higher order variational equations of Hamiltonian systems, In Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 115. doi: 10.1090/conm/549/10850. 
[2] 
T. Combot, Nonintegrability of the equal mass; nbody problem with nonzero angular momentum, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 319340. doi: 10.1007/s105690129417z. 
[3] 
G. Duval and A. J. Maciejewski, Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Annales de l'Institut Fourier, 59 (2009), 28392890. doi: 10.5802/aif.2510. 
[4] 
G. Duval and A. J. Maciejewski, Integrability of Homogeneous potential of degree $k = \pm 2$. An application of higher variational equations, submited, 2012. 
[5] 
J. J. MoralesRuiz and J. P. Ramis, A note on the nonintegrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113120. 
[6] 
E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications Inc., New York, 1960. 
[1] 
Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 45894615. doi: 10.3934/dcds.2014.34.4589 
[2] 
Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree 4. Discrete and Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021228 
[3] 
Regina Martínez, Carles Simó. Nonintegrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 124. doi: 10.3934/dcds.2011.29.1 
[4] 
Mitsuru Shibayama. Nonintegrability criterion for homogeneous Hamiltonian systems via blowingup technique of singularities. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 37073719. doi: 10.3934/dcds.2015.35.3707 
[5] 
Delia Schiera. Existence and nonexistence results for variational higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 51455161. doi: 10.3934/dcds.2018227 
[6] 
Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higherorder LagrangePoincaré equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 309344. doi: 10.3934/dcds.2019013 
[7] 
Dung Le. Higher integrability for gradients of solutions to degenerate parabolic systems. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 597608. doi: 10.3934/dcds.2010.26.597 
[8] 
Sergi Simon. Linearised higher variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 48274854. doi: 10.3934/dcds.2014.34.4827 
[9] 
Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124137. doi: 10.3934/proc.1998.1998.124 
[10] 
R.S. Dahiya, A. Zafer. Oscillation theorems of higher order neutral type differential equations. Conference Publications, 1998, 1998 (Special) : 203219. doi: 10.3934/proc.1998.1998.203 
[11] 
Peiguang Wang, Xiran Wu, Huina Liu. Higher order convergence for a class of set differential equations with initial conditions. Discrete and Continuous Dynamical Systems  S, 2021, 14 (9) : 32333248. doi: 10.3934/dcdss.2020342 
[12] 
Kazuyuki Yagasaki. Higherorder Melnikov method and chaos for twodegreeoffreedom Hamiltonian systems with saddlecenters. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 387402. doi: 10.3934/dcds.2011.29.387 
[13] 
Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete and Continuous Dynamical Systems  B, 2016, 21 (2) : 557573. doi: 10.3934/dcdsb.2016.21.557 
[14] 
Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete and Continuous Dynamical Systems  B, 2015, 20 (8) : 26572661. doi: 10.3934/dcdsb.2015.20.2657 
[15] 
Eduardo Martínez. Higherorder variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81108. doi: 10.3934/jgm.2015.7.81 
[16] 
Chiara Leone, Anna Verde, Giovanni Pisante. Higher integrability results for non smooth parabolic systems: The subquadratic case. Discrete and Continuous Dynamical Systems  B, 2009, 11 (1) : 177190. doi: 10.3934/dcdsb.2009.11.177 
[17] 
Kristian Moring, Christoph Scheven, Sebastian Schwarzacher, Thomas Singer. Global higher integrability of weak solutions of porous medium systems. Communications on Pure and Applied Analysis, 2020, 19 (3) : 16971745. doi: 10.3934/cpaa.2020069 
[18] 
Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869898. doi: 10.3934/ipi.2016025 
[19] 
Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order HamiltonJacobiBellman equations. Approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 39333964. doi: 10.3934/dcds.2015.35.3933 
[20] 
Yu Guo, XiaoBao Shu, Qianbao Yin. Existence of solutions for firstorder Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021236 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]