Ground states of nonlinear Schrödinger systems
Authors:
Jinyong Chang and Zhaoli Liu
Journal:
Proc. Amer. Math. Soc. 138 (2010), 687693
MSC (2000):
Primary 35J10, 35J50, 58E05
Published electronically:
October 2, 2009
MathSciNet review:
2557185
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper concerns the existence of positive radial ground states of the timeindependent Schrödinger system where , and for , and . A result from Sirakov, Comm. Math. Phys. 271 (2007), 199221, is improved.
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Ambrosetti and Eduardo
Colorado, Bound and ground states of coupled nonlinear
Schrödinger equations, C. R. Math. Acad. Sci. Paris
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English and French summaries). MR 2214594
(2006j:35057), 10.1016/j.crma.2006.01.024
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Ambrosetti and Eduardo
Colorado, Standing waves of some coupled nonlinear Schrödinger
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no. 1, 67–82. MR 2302730
(2008f:35369), 10.1112/jlms/jdl020
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Thomas
Bartsch and ZhiQiang
Wang, Note on ground states of nonlinear Schrödinger
systems, J. Partial Differential Equations 19 (2006),
no. 3, 200–207. MR 2252973
(2007h:35274)
 5.
D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Theory of incoherent selffocusing in biased photorefractive media, Phys. Rev. Lett., 78 (1997), 646649.
 6.
E.
N. Dancer and Juncheng
Wei, Spike solutions in coupled nonlinear
Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1189–1208. MR 2457395
(2009m:35102), 10.1090/S0002994708047351
 7.
Djairo
G. de Figueiredo and Orlando
Lopes, Solitary waves for some nonlinear Schrödinger
systems, Ann. Inst. H. Poincaré Anal. Non Linéaire
25 (2008), no. 1, 149–161 (English, with
English and French summaries). MR 2383083
(2009e:35073), 10.1016/j.anihpc.2006.11.006
 8.
B. D. Esry, C. H. Greene, J. P. Burke Jr., and J. L. Bohn, HartreeFock theory for double condensates, Phys. Rev. Lett., 78 (1997), 35943597.
 9.
G. M. Genkin, Modification of superfluidity in a resonantly strongly driven BoseEinstein condensate, Phys. Rev. A, 65 (2002), No. 035604.
 10.
F. T. Hioe, Solitary waves for coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 11521155.
 11.
F.
T. Hioe and Thom
S. Salter, Special set and solutions of coupled nonlinear
Schrödinger equations, J. Phys. A 35 (2002),
no. 42, 8913–8928. MR 1946865
(2003k:35230), 10.1088/03054470/35/42/303
 12.
T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), 50435046.
 13.
TaiChia
Lin and Juncheng
Wei, Ground state of 𝑁 coupled nonlinear Schrödinger
equations in 𝐑ⁿ, 𝐧≤3, Comm. Math. Phys.
255 (2005), no. 3, 629–653. MR 2135447
(2006g:35044), 10.1007/s002200051313x
 14.
TaiChia
Lin and Juncheng
Wei, Spikes in two coupled nonlinear Schrödinger
equations, Ann. Inst. H. Poincaré Anal. Non Linéaire
22 (2005), no. 4, 403–439 (English, with
English and French summaries). MR 2145720
(2006a:35065), 10.1016/j.anihpc.2004.03.004
 15.
TaiChia
Lin and Juncheng
Wei, Spikes in twocomponent systems of nonlinear Schrödinger
equations with trapping potentials, J. Differential Equations
229 (2006), no. 2, 538–569. MR 2263567
(2007h:58031), 10.1016/j.jde.2005.12.011
 16.
Zhaoli
Liu and ZhiQiang
Wang, Multiple bound states of nonlinear Schrödinger
systems, Comm. Math. Phys. 282 (2008), no. 3,
721–731. MR 2426142
(2009k:58022), 10.1007/s002200080546x
 17.
Z.L. Liu and Z.Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, to appear.
 18.
L.
A. Maia, E.
Montefusco, and B.
Pellacci, Positive solutions for a weakly coupled nonlinear
Schrödinger system, J. Differential Equations
229 (2006), no. 2, 743–767. MR 2263573
(2007h:35070), 10.1016/j.jde.2006.07.002
 19.
Eugenio
Montefusco, Benedetta
Pellacci, and Marco
Squassina, Semiclassical states for weakly coupled nonlinear
Schrödinger systems, J. Eur. Math. Soc. (JEMS)
10 (2008), no. 1, 47–71. MR 2349896
(2008i:35088), 10.4171/JEMS/103
 20.
M. Mitchell, Z. Chen, M. Shih, and M. Segev, Selftrapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996) 490493.
 21.
Alessio
Pomponio, Coupled nonlinear Schrödinger systems with
potentials, J. Differential Equations 227 (2006),
no. 1, 258–281. MR 2233961
(2007e:35263), 10.1016/j.jde.2005.09.002
 22.
Boyan
Sirakov, Least energy solitary waves for a system of nonlinear
Schrödinger equations in ℝⁿ, Comm. Math. Phys.
271 (2007), no. 1, 199–221. MR 2283958
(2007k:35477), 10.1007/s002200060179x
 23.
E. Timmermans, Phase separation of BoseEinstein condensates, Phys. Rev. Lett., 81 (1998), 57185721.
 24.
J.C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., to appear.
 25.
Juncheng
Wei and Tobias
Weth, Radial solutions and phase separation in a system of two
coupled Schrödinger equations, Arch. Ration. Mech. Anal.
190 (2008), no. 1, 83–106. MR 2434901
(2009k:35059), 10.1007/s0020500801219
 1.
 N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 26612664.
 2.
 A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453458. MR 2214594 (2006j:35057)
 3.
 A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc. (2), 75 (2007), 6782. MR 2302730 (2008f:35369)
 4.
 T. Bartsch and Z.Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200207. MR 2252973 (2007h:35274)
 5.
 D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Theory of incoherent selffocusing in biased photorefractive media, Phys. Rev. Lett., 78 (1997), 646649.
 6.
 E. N. Dancer and J.C. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 11891208. MR 2457395
 7.
 D.G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 149161. MR 2383083 (2009e:35073)
 8.
 B. D. Esry, C. H. Greene, J. P. Burke Jr., and J. L. Bohn, HartreeFock theory for double condensates, Phys. Rev. Lett., 78 (1997), 35943597.
 9.
 G. M. Genkin, Modification of superfluidity in a resonantly strongly driven BoseEinstein condensate, Phys. Rev. A, 65 (2002), No. 035604.
 10.
 F. T. Hioe, Solitary waves for coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 11521155.
 11.
 F. T. Hioe and T. S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 89138928. MR 1946865 (2003k:35230)
 12.
 T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), 50435046.
 13.
 T.C. Lin and J.C. Wei, Ground state of coupled nonlinear Schrödinger equations in , , Comm. Math. Phys., 255 (2005), 629653. MR 2135447 (2006g:35044)
 14.
 T.C. Lin and J.C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403439. MR 2145720 (2006a:35065)
 15.
 T.C. Lin and J.C. Wei, Spikes in twocomponent systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538569. MR 2263567 (2007h:58031)
 16.
 Z.L. Liu and Z.Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721731. MR 2426142
 17.
 Z.L. Liu and Z.Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, to appear.
 18.
 L.A. Maia, E. Montefusco, and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743767. MR 2263573 (2007h:35070)
 19.
 E. Montefusco, B. Pellacci, and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 4771. MR 2349896 (2008i:35088)
 20.
 M. Mitchell, Z. Chen, M. Shih, and M. Segev, Selftrapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996) 490493.
 21.
 A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258281. MR 2233961 (2007e:35263)
 22.
 B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in , Comm. Math. Phys., 271 (2007), 199221. MR 2283958 (2007k:35477)
 23.
 E. Timmermans, Phase separation of BoseEinstein condensates, Phys. Rev. Lett., 81 (1998), 57185721.
 24.
 J.C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., to appear.
 25.
 J.C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83106. MR 2434901
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Additional Information
Jinyong Chang
Affiliation:
(J. Chang and Z. Liu) School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China;
(J. Chang) Department of Mathematics, Changzhi University, Shanxi 046011, People’s Republic of China
Zhaoli Liu
Affiliation:
(J. Chang and Z. Liu) School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China
DOI:
http://dx.doi.org/10.1090/S0002993909100904
Keywords:
Schr\"odinger system,
nontrivial ground state,
Morse index.
Received by editor(s):
March 24, 2009
Received by editor(s) in revised form:
June 19, 2009
Published electronically:
October 2, 2009
Additional Notes:
This work was supported by NSFC (10825106)
Communicated by:
Yingfei Yi
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
