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December  2016, 9(6): 1775-1795. doi: 10.3934/dcdss.2016074

Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere

Ping-Liang Huang 1, and  Youde Wang 2,

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

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper,we consider the so called generalized inhomogeneous Schrödinger flows from a closed Riemann surface $M$ into the standard 2-sphere $S^2$ associated with the energy functional given by \begin{align*} E_{f,P}(u)=\int_M\left(\frac{1}{2}f|\nabla u|^2+P(u_3)\right)dV_g. \end{align*} We showed the existence of special periodic solutions to the generalized inhomogeneous Schrödinger flows from $M$ with convolution symmetry (especially $M = S^2$) into $S^2$ when the function $f$ and $P$ satisfy certain conditions respectively. Especially, we show that the inhomogeneous Heisenberg spin chain system from a closed Riemann surface with convolution symmetry admits some special periodic solutions if the coupling function $f$ satisfies some suitable conditions. We also prove that there exist an infinite number of special periodic solutions to the Landau-Lifshitz system with an external magnetic field from $S^2$ into $S^2$.
Keywords: convolution symmtry., special periodic solution, $S^1$-invariant, inhomogeneous Heisenberg spin chain system, Schrödinger flow.
Mathematics Subject Classification: Primary: 58J60, 35Q55; Secondary: 37K2.
Citation: Ping-Liang Huang, Youde Wang. Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1775-1795. doi: 10.3934/dcdss.2016074
References:
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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.  Google Scholar

[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.  Google Scholar

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Q. Ding, A note on NLS and the Schödinger flow of maps, Phys. Lett. A, 248 (1998), 49-54. Google Scholar

[4]

W. Ding, Lusternik-Schnirelmann theory for harmonic maps, Acta Math. Sinica (N. S.), 2 (1986), 105-122. doi: 10.1007/BF02564873.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. L. Huang, On some inhomogeneous Geometirc PDEs, Ph.D thesis, AMSS. CAS. in Beijing, 2007. Google Scholar

[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.  Google Scholar

[12]

S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[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.  Google Scholar

[14]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Struwe, Variational Methods, $3^{rd}$ edition, Springer-Verlag, Berlin Heidelberg, 2000. doi: 10.1007/978-3-662-04194-9.  Google Scholar

[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.  Google Scholar

[22]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24. doi: 10.2307/1971131.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Q. Ding, A note on NLS and the Schödinger flow of maps, Phys. Lett. A, 248 (1998), 49-54. Google Scholar

[4]

W. Ding, Lusternik-Schnirelmann theory for harmonic maps, Acta Math. Sinica (N. S.), 2 (1986), 105-122. doi: 10.1007/BF02564873.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. L. Huang, On some inhomogeneous Geometirc PDEs, Ph.D thesis, AMSS. CAS. in Beijing, 2007. Google Scholar

[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.  Google Scholar

[12]

S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[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.  Google Scholar

[14]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Struwe, Variational Methods, $3^{rd}$ edition, Springer-Verlag, Berlin Heidelberg, 2000. doi: 10.1007/978-3-662-04194-9.  Google Scholar

[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.  Google Scholar

[22]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24. doi: 10.2307/1971131.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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