Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Ground states of nonlinear Schrödinger systems

Author(s): Jinyong Chang; Zhaoli Liu
Journal: Proc. Amer. Math. Soc. 138 (2010), 687-693.
MSC (2000): Primary 35J10, 35J50, 58E05
Posted: October 2, 2009
MathSciNet review: 2557185
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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

$\displaystyle \left\{\begin{array}{ll} -\Delta{u_1}+\lambda_1u_1=\mu_1u_1^3+\be... ...1(x)\to0, u_2(x)\to0, &\text{as} \vert x\vert\to\infty, \end{array}\right.$

where

, $\lambda_j>0$ and $\mu_j>0$ for

, and $\beta>0$ . A result from Sirakov, Comm. Math. Phys. 271 (2007), 199-221, is improved.

References:

1.: N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
2.: A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. 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), 67-82. 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), 200-207. MR 2252973 (2007h:35274)
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 J.-C. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc., 361 (2009), 1189-1208. 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), 149-161. MR 2383083 (2009e:35073)
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 T. S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928. 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), 5043-5046.
13.: T.-C. Lin and J.C. Wei, Ground state of coupled nonlinear Schrödinger equations in , $n\leq 3$ , Comm. Math. Phys., 255 (2005), 629-653. 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), 403-439. MR 2145720 (2006a:35065)
15.: T.-C. Lin and J.C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569. MR 2263567 (2007h:58031)
16.: Z.L. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. 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), 743-767. 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), 47-71. MR 2349896 (2008i:35088)
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.: A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281. MR 2233961 (2007e:35263)
22.: B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}\sp n$ , Comm. Math. Phys., 271 (2007), 199-221. MR 2283958 (2007k:35477)
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.: 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), 83-106. MR 2434901

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|