×

Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three. (English) Zbl 1322.35122

The article is concerned with stability questions for a 3D Schrödinger equation describing a delta-type point interaction of a strength which depends on the wave function according to a power-type law, implying that the equation is nonlinear. The authors first prove the existence of standing wave solutions with a spatially localized amplitude function (which is computable in closed form). The main goal is then to investigate the stability of these standing waves. The authors prove orbital stability if the exponent in the considered power law is between 0 and 1, and orbital instability if it is larger than or equal to 1. Moreover, they prove asymptotic stability (in an appropriate sense) if the exponent is less than \(1/\sqrt{2}\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J10 Schrödinger operator, Schrödinger equation
35B35 Stability in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] DOI: 10.1016/S0294-1449(02)00022-7 · Zbl 1028.35137
[2] DOI: 10.1016/j.anihpc.2003.01.002 · Zbl 1042.35070
[3] Albeverio S., Solvable Models in Quantum Mechanics (2005)
[4] DOI: 10.1080/03605300801970937 · Zbl 1185.35247
[5] Buslaev V. S., St. Petersb. Math. J. 4 pp 1111– (1993)
[6] DOI: 10.1090/trans2/164/04
[7] DOI: 10.1016/S0294-1449(02)00018-5 · Zbl 1028.35139
[8] DOI: 10.1002/cpa.20052
[9] DOI: 10.1002/cpa.20052
[10] DOI: 10.1007/s00220-008-0605-3 · Zbl 1155.35092
[11] DOI: 10.1002/mma.682 · Zbl 1095.35038
[12] DOI: 10.1088/0305-4470/38/22/022 · Zbl 1085.81039
[13] DOI: 10.1103/PhysRevA.83.033828
[14] DOI: 10.1016/j.jfa.2009.08.010 · Zbl 1187.35238
[15] DOI: 10.1016/S0167-2789(02)00626-7 · Zbl 1098.74614
[16] DOI: 10.1007/s00220-004-1128-1 · Zbl 1075.35075
[17] DOI: 10.1142/S0129055X05002522 · Zbl 1086.82013
[18] DOI: 10.1016/j.aim.2007.04.018 · Zbl 1126.35065
[19] DOI: 10.3934/dcds.2008.21.137 · Zbl 1154.35082
[20] Gradshteyn I. S., Tables of Integrals, Series and Products (1965)
[21] DOI: 10.1016/0022-1236(90)90016-E · Zbl 0711.58013
[22] DOI: 10.1155/S1073792804132340 · Zbl 1072.35167
[23] DOI: 10.3934/cpaa.2012.11.1063 · Zbl 1282.35352
[24] DOI: 10.1103/PhysRevB.47.10402
[25] Reed M., Methods of Modern Mathematical Physics. IV. Analysis of Operators (1977)
[26] DOI: 10.1007/BF02096557 · Zbl 0721.35082
[27] DOI: 10.1016/0022-0396(92)90098-8 · Zbl 0795.35073
[28] DOI: 10.1103/PhysRevE.63.036601
[29] DOI: 10.1002/cpa.3012 · Zbl 1031.35137
[30] DOI: 10.1155/S1073792802201063 · Zbl 1011.35120
[31] DOI: 10.1002/cpa.3160390103 · Zbl 0594.35005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.