Ground states of nonlinear Schrödinger systems

Authors:
Jinyong Chang and Zhaoli Liu

Journal:
Proc. Amer. Math. Soc. **138** (2010), 687-693

MSC (2000):
Primary 35J10, 35J50, 58E05

Published electronically:
October 2, 2009

MathSciNet review:
2557185

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns the existence of positive radial ground states of the time-independent Schrödinger system

*Comm. Math. Phys.*

**271**(2007), 199-221, is improved.

**1.**N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background,*Phys. Rev. Lett.*,**82**(1999), 2661-2664.**2.**Antonio Ambrosetti and Eduardo Colorado,*Bound and ground states of coupled nonlinear Schrödinger equations*, C. R. Math. Acad. Sci. Paris**342**(2006), no. 7, 453–458 (English, with English and French summaries). MR**2214594**, 10.1016/j.crma.2006.01.024**3.**Antonio Ambrosetti and Eduardo Colorado,*Standing waves of some coupled nonlinear Schrödinger equations*, J. Lond. Math. Soc. (2)**75**(2007), no. 1, 67–82. MR**2302730**, 10.1112/jlms/jdl020**4.**Thomas Bartsch and Zhi-Qiang Wang,*Note on ground states of nonlinear Schrödinger systems*, J. Partial Differential Equations**19**(2006), no. 3, 200–207. MR**2252973****5.**D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Theory of incoherent self-focusing in biased photorefractive media,*Phys. Rev. Lett.*,**78**(1997), 646-649.**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**, 10.1090/S0002-9947-08-04735-1**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**, 10.1016/j.anihpc.2006.11.006**8.**B. D. Esry, C. H. Greene, J. P. Burke Jr., and J. L. Bohn, Hartree-Fock theory for double condensates,*Phys. Rev. Lett.*,**78**(1997), 3594-3597.**9.**G. M. Genkin, Modification of superfluidity in a resonantly strongly driven Bose-Einstein condensate,*Phys. Rev. A*,**65**(2002), No. 035604.**10.**F. T. Hioe, Solitary waves for coupled nonlinear Schrödinger equations,*Phys. Rev. Lett.*,**82**(1999), 1152-1155.**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**, 10.1088/0305-4470/35/42/303**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), 5043-5046.**13.**Tai-Chia 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**, 10.1007/s00220-005-1313-x**14.**Tai-Chia 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**, 10.1016/j.anihpc.2004.03.004**15.**Tai-Chia Lin and Juncheng Wei,*Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials*, J. Differential Equations**229**(2006), no. 2, 538–569. MR**2263567**, 10.1016/j.jde.2005.12.011**16.**Zhaoli Liu and Zhi-Qiang Wang,*Multiple bound states of nonlinear Schrödinger systems*, Comm. Math. Phys.**282**(2008), no. 3, 721–731. MR**2426142**, 10.1007/s00220-008-0546-x**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**, 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**, 10.4171/JEMS/103**20.**M. Mitchell, Z. Chen, M. Shih, and M. Segev, Self-trapping of partially spatially incoherent light,*Phys. Rev. Lett.*,**77**(1996) 490-493.**21.**Alessio Pomponio,*Coupled nonlinear Schrödinger systems with potentials*, J. Differential Equations**227**(2006), no. 1, 258–281. MR**2233961**, 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**, 10.1007/s00220-006-0179-x**23.**E. Timmermans, Phase separation of Bose-Einstein condensates,*Phys. Rev. Lett.*,**81**(1998), 5718-5721.**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**, 10.1007/s00205-008-0121-9

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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/S0002-9939-09-10090-4

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.