Skip to main content
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
/content/aip/journal/jmp/57/8/10.1063/1.4960045
1.
Adams, D. R. , Sobolev Spaces (Academic Press, 1975).
2.
Akrivis, G. D. , Dougalis, V. A. , Karakashian, O. A. , and Mckinney, W. R. , Numerical Approximation of Singular Solution of the Damped Nonlinear Schrödinger Equation, ENUMATH 97 (Heidelberg) (World Scientific, River Edge, NJ, 1998), pp. 117124.
3.
Barashenkov, I. V. , Alexeeva, N. V. , and Zemlianaya, E. V. , “Two and three dimensional oscillons in nonlinear Faradey resonance,” Phys. Rev. Lett. 89(10), 104101 (2002).
http://dx.doi.org/10.1103/PhysRevLett.89.104101
4.
Bourgain, J. , “Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case,” J. Am. Math. Soc. 12(1), 145171 (1991).
http://dx.doi.org/10.1090/S0894-0347-99-00283-0
5.
Bourgain, J. , Global Solutions of Nonlinear Schrödinger Equation (American Mathematical Society Colloquium Publications, Providence, RI, 1991), Vol. 46.
6.
Cazenave, T. , Semilinear Schrödinger Equations, Lecture Notes in Mathematics (Courant Institute of Mathematical Sciences, New York University, New York, 2003), Vol.10.
7.
Chen, J. , “On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient,” Czech Math. J. 60(3), 715736 (2012).
http://dx.doi.org/10.1007/s10587-010-0046-y
8.
Chen, J. and Guo, B. , “Sharp constant of improved Gagliardo-Nirenberg inequality and its application,” Ann. Mat. 190, 341354 (2011).
http://dx.doi.org/10.1007/s10231-010-0152-3
9.
Cho, Y. , Hajaiej, H. , Hwang, G. , and Ozawa, T. , “On the orbital stability of fractional Schrödinger equations,” Commun. Pure Appl. Anal. 13(3), 12671282 (2014).
http://dx.doi.org/10.3934/cpaa.2014.13.1267
10.
Cho, Y. , Hwang, G. , Kwon, S. , and Lee, S. , “Profile decompositions and blowup phenomena of mass critical fractional Schrödinger equations,” Nonlinear Anal. 8(6), 1229 (2013).
http://dx.doi.org/10.1016/j.na.2013.03.002
11.
Cho, Y. , Hwang, G. , Kwon, S. , and Lee, S. , “On the finite time blowup for mass-critical Hartree equations,” Proc. R. Soc. Edinburgh, Sect. A: Math. 145(03), 467479 (2015).
http://dx.doi.org/10.1017/S030821051300142X
12.
Cho, Y. and Ozawa, T. , “Sobolev inequalities with symmetry,” Commun. Contemp. Math. 11, 355365 (2009).
http://dx.doi.org/10.1142/S0219199709003399
13.
Fibich, G. and Wang, X. P. , “Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearity,” Physica D 175, 96108 (2003).
http://dx.doi.org/10.1016/S0167-2789(02)00626-7
14.
Frank, R. L. and Lenzmann, E. , “Uniqueness and nondegeneracy of ground states for (Δ)sQ + QQα+1 = 0 in ℝ,” e-print arXiv:1009.4042v2 [math.AP] (2015).
15.
Frank, R. L. , Lenzmann, E. , and Silvestre, L. , “Uniqueness of radial solutions for the fractional Laplacian,” Commun. Pure Appl. Math. 69(9), 16711726 (2016).
http://dx.doi.org/10.1002/cpa.21591
16.
Fukuizumi, R. and Ohta, M. , “Instability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities,” J. Math. Kyoto Univ. 45, 145158 (2005).
17.
Gill, T. S. , “Optical guiding of laser beam in nonuniform plasma,” Pramana J. Phys. 55, 845852 (2000).
18.
Guo, B. , Han, Y. , and Xin, J. , “Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation,” Appl. Math. Comput. 204(1), 468477 (2008).
http://dx.doi.org/10.1016/j.amc.2008.07.003
19.
Guo, B. and Huang, D. , “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
http://dx.doi.org/10.1063/1.4746806
20.
Guo, B. and Huo, Z. , “Global well-posedness for the fractional nonlinear Schrödinger equation,” Commun. Partial Differ. Equations 36(2), 247255 (2010).
http://dx.doi.org/10.1080/03605302.2010.503769
21.
Guo, B. and Huo, Z. , “Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation,” Fractional Calculus Appl. Anal. 16(1), 226242 (2013).
http://dx.doi.org/10.2478/s13540-013-0014-y
22.
Guo, Z. , Sire, Y. , Wang, Y. , and Zhao, L. , “On the energy-critical fractional Schrödinger equation in the radial case,” e-print arXiv:1310.6816v1 [math.AP] (2013).
23.
Guo, Z. and Wang, Y. , “Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations,” J. Anal. Math. 124(1), 138 (2014).
http://dx.doi.org/10.1007/s11854-014-0025-6
24.
Hajaiej, H. , Molinet, L. , Ozawa, T. , and Wan, B. , “Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,” RIMS Kokyuroku Bessatsu B26, 159199 (2011).
25.
Hezzi, H. , Marzouk, A. , and Saanouni, T. , “A note on the inhomogeneous Schrödinger equation with mixed power nonlinearity,” Commun. Math. Anal. 18(2), 3655 (2015).
26.
Kavian, O. and Weissler, F. B. , “Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation,” Mich. Math. J. 41(1), 151173 (1994).
http://dx.doi.org/10.1307/mmj/1029004922
27.
Laskin, N. , “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298304 (2000).
http://dx.doi.org/10.1016/S0375-9601(00)00201-2
28.
Laskin, N. , “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
http://dx.doi.org/10.1103/PhysRevE.66.056108
29.
Le Coz, S. , “A note on Berestycki-Cazenave classical instability result for nonlinear Schrödinger equations,” Adv. Nonlinear Stud. 8(3), 455463 (2008).
http://dx.doi.org/10.1515/ans-2008-0302
30.
Lions, P. L. , “Symetrie et compacité dans les espaces de Sobolev,” J. Funct. Anal. 49, 315334 (1982).
http://dx.doi.org/10.1016/0022-1236(82)90072-6
31.
Liu, C. S. and Tripathi, V. K. , “Laser guiding in an axially nonuniform plasma channel,” Phys. Plasmas 1, 31003103 (1994).
http://dx.doi.org/10.1063/1.870501
32.
Liu, Y. , Wang, X. P. , and Wang, K. , “Instability of standing waves of the Schrödinger equations with inhomogeneous nonlinearity,” Trans. Am. Math. Soc. 358, 21052122 (2006).
http://dx.doi.org/10.1090/S0002-9947-05-03763-3
33.
Merle, F. , “Nonexistence of minimal blow up solutions of equations in ℝN,” Ann. Inst. Henri Poincare, Sect. A 64, 3385 (1996).
34.
Payne, L. E. and Sattinger, D. H. , “Saddle points and instability of nonlinear hyperbolic equations,” Isr. J. Math. 22, 273303 (1975).
http://dx.doi.org/10.1007/BF02761595
35.
Strauss, W. , “Existence of solitary waves in higher dimensions,” Commun. Math. Phys. 55(2), 149162 (1977).
http://dx.doi.org/10.1007/BF01626517
36.
Tsurumi, T. and Waditi, M. , “Collapses of wave functions in multidimensional nonlinear Schrödinger equations under harmonic potential,” J. Phys. Soc. Jpn. 66, 30313034 (1997).
http://dx.doi.org/10.1143/JPSJ.66.3031
http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/8/10.1063/1.4960045
Loading
/content/aip/journal/jmp/57/8/10.1063/1.4960045
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/jmp/57/8/10.1063/1.4960045
2016-08-05
2016-09-28

Abstract

Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated.

Loading

Full text loading...

/deliver/fulltext/aip/journal/jmp/57/8/1.4960045.html;jsessionid=rBoUfDN8CWfskvI07f6ToSWF.x-aip-live-03?itemId=/content/aip/journal/jmp/57/8/10.1063/1.4960045&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jmp
true
true

Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
/content/realmedia?fmt=ahah&adPositionList=
&advertTargetUrl=//oascentral.aip.org/RealMedia/ads/&sitePageValue=jmp.aip.org/57/8/10.1063/1.4960045&pageURL=http://scitation.aip.org/content/aip/journal/jmp/57/8/10.1063/1.4960045'
Right1,Right2,Right3,