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Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations
Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem
1. | Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China |
2. | School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, Hebei, China |
We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [
References:
[1] |
W. Ballmann, G. Thorbergsson and W. Ziller,
Closed geodesics on positively curved manifolds, Ann. of Math., 116 (1982), 213-247.
doi: 10.2307/2007062. |
[2] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[3] |
P. P. Hallen and D. N. Rana,
The Existence and stability of equilibrium points in the Robe's restricted three-body problem, Celest. Mech. Dyn. Astr., 79 (2001), 145-155.
doi: 10.1023/A:1011173320720. |
[4] |
X. Hu, Y. Long and S. Sun,
Linear stability of elliptic Lagrangian solutions of the classical planar three-body problem via index theory, Arch. Rational. Mech. Anal., 213 (2014), 993-1045.
doi: 10.1007/s00205-014-0749-6. |
[5] |
X. Hu and Y. Ou,
Collision index and stability of elliptic relative equilibria in planar $n$-body problem, Commun. Math. Phys., 348 (2016), 803-845.
doi: 10.1007/s00220-016-2695-7. |
[6] |
X. Hu and S. Sun,
Index and stability of symmetric periodic orbits in Hamiltonian systems with its application to figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[7] |
X. Hu and S. Sun,
Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223 (2010), 98-119.
doi: 10.1016/j.aim.2009.07.017. |
[8] |
Y. Long,
The structure of the singular symplectic matrix set, Sci. China. Ser. A. (English Ed.), 34 (1991), 897-907.
|
[9] |
Y. Long,
Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.
doi: 10.2140/pjm.1999.187.113. |
[10] |
Y. Long,
Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.
doi: 10.1006/aima.2000.1914. |
[11] |
Y. Long,
Index Theory for Symplectic Paths with Applications Progress in Math. 207, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[12] |
A. R. Plastino and A. Plastino,
Robe's restricted three-body problem revisited, Celest. Mech. Dyn. Astr., 61 (1995), 197-206.
doi: 10.1007/BF00048515. |
[13] |
H. A. G. Robe,
A new kind of three-body problem, Celest. Mech., 16 (1977), 343-351.
doi: 10.1007/BF01232659. |
[14] |
A. K. Shrivastava and D. Garain,
Effect of perturbation on the location of liberation point in the Robe restricted problem of three bodies, Celest. Mech., 51 (1991), 67-73.
doi: 10.1007/BF02426670. |
[15] |
K. T. Singh, B. S. Kushvah and B. Ishwar,
Stability of triangular equilibrium points in Robe's generalised restricted three body problem, in Celestial Mechanics: Recent Trends (eds. B.Ishwar), Narosa Publishing House Pvt. Ltd., New Delhi, India, (2006), 65-70.
|
[16] |
Q. Zhou and Y. Long,
Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems, J. Diff. Equa., 258 (2015), 3851-3879.
doi: 10.1016/j.jde.2015.01.045. |
[17] |
Q. Zhou and Y. Long, Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem, preprint, arXiv: 1510.06822. |
[18] |
Q. Zhou and Y. Long,
The reduction on the linear stability of elliptic Euler-Moulton solutions of the $n$-body problem to those of $3$-body problems, Celest. Mech. Dyn. Astr., (2016), 1-32.
doi: 10.1007/s10569-016-9732-x. |
show all references
References:
[1] |
W. Ballmann, G. Thorbergsson and W. Ziller,
Closed geodesics on positively curved manifolds, Ann. of Math., 116 (1982), 213-247.
doi: 10.2307/2007062. |
[2] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3. |
[3] |
P. P. Hallen and D. N. Rana,
The Existence and stability of equilibrium points in the Robe's restricted three-body problem, Celest. Mech. Dyn. Astr., 79 (2001), 145-155.
doi: 10.1023/A:1011173320720. |
[4] |
X. Hu, Y. Long and S. Sun,
Linear stability of elliptic Lagrangian solutions of the classical planar three-body problem via index theory, Arch. Rational. Mech. Anal., 213 (2014), 993-1045.
doi: 10.1007/s00205-014-0749-6. |
[5] |
X. Hu and Y. Ou,
Collision index and stability of elliptic relative equilibria in planar $n$-body problem, Commun. Math. Phys., 348 (2016), 803-845.
doi: 10.1007/s00220-016-2695-7. |
[6] |
X. Hu and S. Sun,
Index and stability of symmetric periodic orbits in Hamiltonian systems with its application to figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[7] |
X. Hu and S. Sun,
Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223 (2010), 98-119.
doi: 10.1016/j.aim.2009.07.017. |
[8] |
Y. Long,
The structure of the singular symplectic matrix set, Sci. China. Ser. A. (English Ed.), 34 (1991), 897-907.
|
[9] |
Y. Long,
Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149.
doi: 10.2140/pjm.1999.187.113. |
[10] |
Y. Long,
Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.
doi: 10.1006/aima.2000.1914. |
[11] |
Y. Long,
Index Theory for Symplectic Paths with Applications Progress in Math. 207, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[12] |
A. R. Plastino and A. Plastino,
Robe's restricted three-body problem revisited, Celest. Mech. Dyn. Astr., 61 (1995), 197-206.
doi: 10.1007/BF00048515. |
[13] |
H. A. G. Robe,
A new kind of three-body problem, Celest. Mech., 16 (1977), 343-351.
doi: 10.1007/BF01232659. |
[14] |
A. K. Shrivastava and D. Garain,
Effect of perturbation on the location of liberation point in the Robe restricted problem of three bodies, Celest. Mech., 51 (1991), 67-73.
doi: 10.1007/BF02426670. |
[15] |
K. T. Singh, B. S. Kushvah and B. Ishwar,
Stability of triangular equilibrium points in Robe's generalised restricted three body problem, in Celestial Mechanics: Recent Trends (eds. B.Ishwar), Narosa Publishing House Pvt. Ltd., New Delhi, India, (2006), 65-70.
|
[16] |
Q. Zhou and Y. Long,
Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems, J. Diff. Equa., 258 (2015), 3851-3879.
doi: 10.1016/j.jde.2015.01.045. |
[17] |
Q. Zhou and Y. Long, Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem, preprint, arXiv: 1510.06822. |
[18] |
Q. Zhou and Y. Long,
The reduction on the linear stability of elliptic Euler-Moulton solutions of the $n$-body problem to those of $3$-body problems, Celest. Mech. Dyn. Astr., (2016), 1-32.
doi: 10.1007/s10569-016-9732-x. |


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