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Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere
1. | Department of Mathematics, Shanghai University, Shanghai 200444, China |
2. | Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China |
References:
[1] |
W. Chen and J. Jost, Maps with prescribed tension fields, Comm. Anal. Geom., 12 (2004), 93-109.
doi: 10.4310/CAG.2004.v12.n1.a6. |
[2] |
N. Chang, J. Shatah and K. Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math., 53 (2000), 590-602.
doi: 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R. |
[3] |
Q. Ding, A note on NLS and the Schödinger flow of maps, Phys. Lett. A, 248 (1998), 49-54. |
[4] |
W. Ding, Lusternik-Schnirelmann theory for harmonic maps, Acta Math. Sinica (N. S.), 2 (1986), 105-122.
doi: 10.1007/BF02564873. |
[5] |
W. Y. Ding and Y. D. Wang, Schrödinger flow of mappings into sympletic manifolds, Sci. China Ser. A, 41 (1998), 746-755.
doi: 10.1007/BF02901957. |
[6] |
W. Y. Ding and Y. D. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464.
doi: 10.1007/BF02877074. |
[7] |
W. Y. Ding and H. Yin, Special periodic Solutions of Schrödinger flow, Math. Z., 253 (2006), 555-570.
doi: 10.1007/s00209-005-0922-6. |
[8] |
J. Eells and L. Lemaire, Another report on harmonic maps, London Math Soc., 20 (1988), 385-524.
doi: 10.1112/blms/20.5.385. |
[9] |
S. Gustafson and J. Shatah, The stability of localize solutions of Landau-Lifshitz equations, J. Comm. Pure Appl. Math., 55 (2002), 1136-1159.
doi: 10.1002/cpa.3024. |
[10] |
P. L. Huang, On some inhomogeneous Geometirc PDEs, Ph.D thesis, AMSS. CAS. in Beijing, 2007. |
[11] |
Y. X. Li and Y. D. Wang, Bubbling location for f-harmonic maps and inhomogeneous Landau-Lifshitz equations, Comm. Math. Helv., 81 (2006), 433-448. |
[12] |
S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York-Heidelberg, 1972. |
[13] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.
doi: 10.1016/0370-1573(90)90130-T. |
[14] |
R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[15] |
P. Y. H. Pang, H. Wang and Y. D. Wang, Local existence for inhomogeneous Schrödinger flow into Kähler manifolds, Acta Math. Sinica, English Ser., 16 (2000), 487-504.
doi: 10.1007/s101140000060. |
[16] |
P. Y. H. Pang, H. Wang and Y. D. Wang, Schrödinger flow for maps into Kähler manifolds, Asian J. Math., 5 (2001), 509-533.
doi: 10.4310/AJM.2001.v5.n3.a7. |
[17] |
P. Y. H. Pang, H. Wang and Y. D. Wang, Schrödinger flow on Hermitian locally symmetric spaces, Comm. Anal. Geom., 10 (2002), 653-681.
doi: 10.4310/CAG.2002.v10.n4.a1. |
[18] |
X. Peng and G. Wang, Harmonic maps with a prescribed potential, C. R. Acad. Sci., Paris, Sér. I, Math., 327 (1998), 271-276.
doi: 10.1016/S0764-4442(98)80145-6. |
[19] |
B. Piette and W. J. Zakrzewski, Localized solutions in a two-dimensional Landau-Lifshitz model, Physic D, 119 (1998), 314-326.
doi: 10.1016/S0167-2789(98)00084-0. |
[20] |
M. Struwe, Variational Methods, $3^{rd}$ edition, Springer-Verlag, Berlin Heidelberg, 2000.
doi: 10.1007/978-3-662-04194-9. |
[21] |
P. Sulem, C. Sulem and C. Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys., 107 (1986), 431-454.
doi: 10.1007/BF01220998. |
[22] |
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24.
doi: 10.2307/1971131. |
[23] |
H. Wang and Y. D. Wang, Global existence of inhomogeneous Heisenberg spin systems and Schrödinger flow, Internat. J. Math., 11 (2000), 1079-1114.
doi: 10.1142/S0129167X00000568. |
[24] |
H. Yin, Periodic solutions of Schrödinger flow from $S^3$ to $S^2$, Chinese Ann. Math. Ser. B, 27 (2006), 401-410.
doi: 10.1007/s11401-005-0101-4. |
[25] |
Y. Zhou, B. Guo and S. Tan, Existence and uniqueness of Smooth solution for system of ferromagnetic chain, Science in China A, 34 (1991), 257-266. |
show all references
References:
[1] |
W. Chen and J. Jost, Maps with prescribed tension fields, Comm. Anal. Geom., 12 (2004), 93-109.
doi: 10.4310/CAG.2004.v12.n1.a6. |
[2] |
N. Chang, J. Shatah and K. Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math., 53 (2000), 590-602.
doi: 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R. |
[3] |
Q. Ding, A note on NLS and the Schödinger flow of maps, Phys. Lett. A, 248 (1998), 49-54. |
[4] |
W. Ding, Lusternik-Schnirelmann theory for harmonic maps, Acta Math. Sinica (N. S.), 2 (1986), 105-122.
doi: 10.1007/BF02564873. |
[5] |
W. Y. Ding and Y. D. Wang, Schrödinger flow of mappings into sympletic manifolds, Sci. China Ser. A, 41 (1998), 746-755.
doi: 10.1007/BF02901957. |
[6] |
W. Y. Ding and Y. D. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464.
doi: 10.1007/BF02877074. |
[7] |
W. Y. Ding and H. Yin, Special periodic Solutions of Schrödinger flow, Math. Z., 253 (2006), 555-570.
doi: 10.1007/s00209-005-0922-6. |
[8] |
J. Eells and L. Lemaire, Another report on harmonic maps, London Math Soc., 20 (1988), 385-524.
doi: 10.1112/blms/20.5.385. |
[9] |
S. Gustafson and J. Shatah, The stability of localize solutions of Landau-Lifshitz equations, J. Comm. Pure Appl. Math., 55 (2002), 1136-1159.
doi: 10.1002/cpa.3024. |
[10] |
P. L. Huang, On some inhomogeneous Geometirc PDEs, Ph.D thesis, AMSS. CAS. in Beijing, 2007. |
[11] |
Y. X. Li and Y. D. Wang, Bubbling location for f-harmonic maps and inhomogeneous Landau-Lifshitz equations, Comm. Math. Helv., 81 (2006), 433-448. |
[12] |
S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York-Heidelberg, 1972. |
[13] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.
doi: 10.1016/0370-1573(90)90130-T. |
[14] |
R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[15] |
P. Y. H. Pang, H. Wang and Y. D. Wang, Local existence for inhomogeneous Schrödinger flow into Kähler manifolds, Acta Math. Sinica, English Ser., 16 (2000), 487-504.
doi: 10.1007/s101140000060. |
[16] |
P. Y. H. Pang, H. Wang and Y. D. Wang, Schrödinger flow for maps into Kähler manifolds, Asian J. Math., 5 (2001), 509-533.
doi: 10.4310/AJM.2001.v5.n3.a7. |
[17] |
P. Y. H. Pang, H. Wang and Y. D. Wang, Schrödinger flow on Hermitian locally symmetric spaces, Comm. Anal. Geom., 10 (2002), 653-681.
doi: 10.4310/CAG.2002.v10.n4.a1. |
[18] |
X. Peng and G. Wang, Harmonic maps with a prescribed potential, C. R. Acad. Sci., Paris, Sér. I, Math., 327 (1998), 271-276.
doi: 10.1016/S0764-4442(98)80145-6. |
[19] |
B. Piette and W. J. Zakrzewski, Localized solutions in a two-dimensional Landau-Lifshitz model, Physic D, 119 (1998), 314-326.
doi: 10.1016/S0167-2789(98)00084-0. |
[20] |
M. Struwe, Variational Methods, $3^{rd}$ edition, Springer-Verlag, Berlin Heidelberg, 2000.
doi: 10.1007/978-3-662-04194-9. |
[21] |
P. Sulem, C. Sulem and C. Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys., 107 (1986), 431-454.
doi: 10.1007/BF01220998. |
[22] |
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24.
doi: 10.2307/1971131. |
[23] |
H. Wang and Y. D. Wang, Global existence of inhomogeneous Heisenberg spin systems and Schrödinger flow, Internat. J. Math., 11 (2000), 1079-1114.
doi: 10.1142/S0129167X00000568. |
[24] |
H. Yin, Periodic solutions of Schrödinger flow from $S^3$ to $S^2$, Chinese Ann. Math. Ser. B, 27 (2006), 401-410.
doi: 10.1007/s11401-005-0101-4. |
[25] |
Y. Zhou, B. Guo and S. Tan, Existence and uniqueness of Smooth solution for system of ferromagnetic chain, Science in China A, 34 (1991), 257-266. |
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