Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere

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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

Abstract
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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 and Continuous Dynamical Systems - S, 2016, 9 (6) : 1775-1795. doi: 10.3934/dcdss.2016074

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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