Ground states of nonlinear Schrödinger systems
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- by Jinyong Chang and Zhaoli Liu PDF
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Abstract:
This paper concerns the existence of positive radial ground states of the time-independent Schrödinger system \begin{equation*} \left \{\begin {array}{ll} -\Delta {u_1}+\lambda _1u_1=\mu _1u_1^3+\beta {u_2^2u_1}, \quad &\text {in}\ \mathbb R^n,\\ -\Delta {u_2}+\lambda _2{u_2}=\mu _2u_2^3+\beta {u_1^2u_2}, \quad &\text {in}\ \mathbb R^n,\\ u_1(x)\to 0,\ \ u_2(x)\to 0,\ \ &\text {as}\ |x|\to \infty , \end{array}\right . \end{equation*} where $n=1,2,3$, $\lambda _j>0$ and $\mu _j>0$ for $j=1,2$, and $\beta >0$. A result from Sirakov, Comm. Math. Phys. 271 (2007), 199-221, is improved.References
- N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
- 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, DOI 10.1016/j.crma.2006.01.024
- 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, DOI 10.1112/jlms/jdl020
- 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
- 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.
- 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, DOI 10.1090/S0002-9947-08-04735-1
- Djairo G. de Figueiredo and Orlando Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 1, 149–161 (English, with English and French summaries). MR 2383083, DOI 10.1016/j.anihpc.2006.11.006
- 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.
- G. M. Genkin, Modification of superfluidity in a resonantly strongly driven Bose-Einstein condensate, Phys. Rev. A, 65 (2002), No. 035604.
- F. T. Hioe, Solitary waves for $N$ coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 1152-1155.
- 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, DOI 10.1088/0305-4470/35/42/303
- 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.
- Tai-Chia Lin and Juncheng Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbf R^n$, $n\leq 3$, Comm. Math. Phys. 255 (2005), no. 3, 629–653. MR 2135447, DOI 10.1007/s00220-005-1313-x
- Tai-Chia Lin and Juncheng Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 4, 403–439 (English, with English and French summaries). MR 2145720, DOI 10.1016/j.anihpc.2004.03.004
- 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, DOI 10.1016/j.jde.2005.12.011
- Zhaoli Liu and Zhi-Qiang Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys. 282 (2008), no. 3, 721–731. MR 2426142, DOI 10.1007/s00220-008-0546-x
- Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, to appear.
- 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, DOI 10.1016/j.jde.2006.07.002
- 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, DOI 10.4171/JEMS/103
- M. Mitchell, Z. Chen, M. Shih, and M. Segev, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996) 490-493.
- Alessio Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations 227 (2006), no. 1, 258–281. MR 2233961, DOI 10.1016/j.jde.2005.09.002
- Boyan Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\Bbb R^n$, Comm. Math. Phys. 271 (2007), no. 1, 199–221. MR 2283958, DOI 10.1007/s00220-006-0179-x
- E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
- J.C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., to appear.
- 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, DOI 10.1007/s00205-008-0121-9
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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 687-693
- MSC (2000): Primary 35J10, 35J50, 58E05
- DOI: https://doi.org/10.1090/S0002-9939-09-10090-4
- MathSciNet review: 2557185