×

Unstable normalized standing waves for the space periodic NLS. (English) Zbl 1405.35191

Summary: For the stationary nonlinear Schr\`‘odinger equation \(-\Delta u+ V(x)u- f(u) = \lambda u\) with periodic potential \(V\) we study the existence and stability properties of multibump solutions with prescribed \(L^2\)-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow us to match the \(L^2\)-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated \(L^2\)-sphere, and we show their orbital instability with respect to the Schrödinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] 10.1016/j.jfa.2005.11.010 · Zbl 1126.35057
[2] 10.1515/anona-2016-0123 · Zbl 06992610
[3] 10.1142/S0219199705001763 · Zbl 1070.35083
[4] 10.1142/S0129055X0900361X · Zbl 1171.82304
[5] 10.1512/iumj.1992.41.41052 · Zbl 0796.35043
[6] 10.1090/S0002-9947-09-04669-8 · Zbl 1175.37066
[7] 10.1209/epl/i2003-00579-4
[8] 10.1007/s00013-012-0468-x · Zbl 1260.35098
[9] 10.1016/j.jfa.2017.01.025 · Zbl 06714264
[10] 10.2307/2322987 · Zbl 0565.90007
[11] 10.1007/s00220-017-2866-1 · Zbl 1367.35150
[12] 10.1007/BF01403504 · Zbl 0513.35007
[13] 10.1002/cpa.3160451002 · Zbl 0785.35029
[14] 10.1103/RevModPhys.71.463
[15] 10.1007/978-3-662-00547-7
[16] 10.1080/03605309408821013 · Zbl 0807.35134
[17] 10.1038/nature01452
[18] 10.3934/dcds.2001.7.525 · Zbl 0992.35094
[19] ; Fukuizumi, Differential Integral Equations, 16, 111, (2003)
[20] 10.1002/1522-2616(200011)219:1<109::AID-MANA109>3.3.CO;2-T
[21] 10.1016/0022-1236(87)90044-9 · Zbl 0656.35122
[22] 10.1016/0022-1236(90)90016-E · Zbl 0711.58013
[23] 10.1016/S0294-1449(01)00089-0 · Zbl 1034.35127
[24] 10.2307/2324784 · Zbl 0803.58012
[25] 10.1103/PhysRevA.66.063605
[26] 10.1007/978-1-4612-0741-2
[27] 10.1016/0895-7177(91)90039-A · Zbl 0767.49019
[28] ; Ianni, Adv. Differential Equations, 14, 717, (2009)
[29] 10.1016/S0362-546X(96)00021-1 · Zbl 0877.35091
[30] 10.1007/s00526-010-0374-7 · Zbl 1223.49021
[31] 10.1007/s11784-009-0104-y · Zbl 1189.58004
[32] 10.1137/070683842 · Zbl 1170.35538
[33] 10.1016/S0294-1449(16)30422-X · Zbl 0704.49004
[34] 10.1103/PhysRevA.67.013602
[35] 10.1137/0516004 · Zbl 0581.47049
[36] 10.2307/2154282 · Zbl 0804.35034
[37] 10.1103/RevModPhys.78.179
[38] 10.1215/S0012-7094-93-07004-4 · Zbl 0796.35056
[39] 10.1016/0040-9383(63)90013-2 · Zbl 0122.10702
[40] 10.1090/conm/198/02518
[41] 10.1007/s005260050064 · Zbl 0876.34055
[42] 10.2307/2974858 · Zbl 0832.49015
[43] 10.2307/2323709 · Zbl 0602.49017
[44] 10.1007/s00032-008-0089-9 · Zbl 1179.37101
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.