zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Remarks on the blow-up rate for critical nonlinear Schrödinger equation with harmonic potential. (English) Zbl 1166.35377
Summary: We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a harmonic potential, which models the attractive Bose-Einstein condensate. We establish the sharp lower and upper bounds of blow-up rate as tT (blow-up time), which improve the result of Q. Liu, Y. Zhou and J. Zhang [Appl. Math. Comput. 172, No. 2, 1121–1132 (2006; Zbl 1091.35092)].
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35B40Asymptotic behavior of solutions of PDE
35B45A priori estimates for solutions of PDE
81V80Applications of quantum theory to quantum optics
References:
[1]Bradley, C. C.; Sackett, C. A.; Hulet, R. G.: Bose – Einstein condensation of lithium: observation of limited condensate number, Phys. rev. Lett. 78, 985-989 (1997)
[2]Carles, R.: Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. henri Poincarè 3, 757-772 (2002) · Zbl 1021.81013 · doi:10.1007/s00023-002-8635-4
[3]Carles, R.: Critical nonlinear Schrödinger equations with and without harmonic potential, Math. mod. Meth. appl. Sci. 12, 1513-1523 (2002) · Zbl 1029.35208 · doi:10.1142/S0218202502002215
[4]T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de mètodos màthmatics, 26, IM-UFRJ, Rio de Janeiro, 1993.
[5]Fibich, G.; Merle, F.; Raphaël, P.: Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation, Physica D 220, 1-13 (2006) · Zbl 1100.35097 · doi:10.1016/j.physd.2006.06.010
[6]Kwong, M. K.: Uniqueness of positive solutions of Δu-u+up=0 in RN, Arch. rational mech. Anal. 105, 243-266 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502
[7]Landam, M. J.; Papanicolao, G. C.; Suelm, C.; Sulem, P. L.: Rate of blowup for solutions of nonlinear Schrödinger equation at critical dimension, Phys. rev. A 38, 3837-3843 (1988)
[8]Liu, Q.; Zhou, Y.; Zhang, J.: Upper and lower bound of the blow-up rate for nonlinear Schrödinger equation with a harmonic potential, Appl. math. Comput. 172, 1121-1132 (2006) · Zbl 1091.35092 · doi:10.1016/j.amc.2005.03.011
[9]Merle, F.; Raphaël, P.: Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. funct. Anal. 13, 591-642 (2003) · Zbl 1061.35135 · doi:10.1007/s00039-003-0424-9
[10]Merle, F.; Raphaël, P.: On universality of blow up profile for L2 critical nonlinear Schrödinger equation, Invent. math. 156, 565-672 (2004) · Zbl 1067.35110 · doi:10.1007/s00222-003-0346-z
[11]Merle, F.; Raphaël, P.: Blow up dynamics and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. math. 161, 157-222 (2005) · Zbl 1185.35263 · doi:10.4007/annals.2005.161.157
[12]Merle, F.; Raphaël, P.: On a sharp lower bound on the blow up rate for the L2 critical nonlinear Schrödinger equation, J. am. Soc. 19, 37-90 (2005) · Zbl 1075.35077 · doi:10.1090/S0894-0347-05-00499-6
[13]Oh, Y. G.: Cauchy problem and ehrenfest’s law of nonlinear Schrödinger equations with potentials, J. diff. Eq. 81, 255-274 (1989) · Zbl 0703.35158 · doi:10.1016/0022-0396(89)90123-X
[14]Raphaël, P.: Stability of log – log bound for blow up solutions to the critical nonlinear Schrödinger equation, Math. ann. 331, 577-609 (2005) · Zbl 1082.35143 · doi:10.1007/s00208-004-0596-0
[15]Tsurumi, T.; Wadati, M.: Collapses of wave functions in multidimensional nonlinear Schrödinger equations under harmonic potential, J. phys. Soc. jpn. 66, 3031-3034 (1997) · Zbl 0973.76623 · doi:10.1143/JPSJ.66.3031
[16]Wadati, M.; Tsurumi, T.: Critical number of atoms for the magnetically trapped Bose – Einstein condensate with negative s-wave scattering length, Phys. lett. A 247, 287-293 (1998)
[17]Weinstein, M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. math. Phys. 87, 567-576 (1983) · Zbl 0527.35023 · doi:10.1007/BF01208265
[18]Zhang, J.: Stability of attractive Bose – Einstein condensate, J. statist. Phys. 101, 731-746 (2000) · Zbl 0989.82024 · doi:10.1023/A:1026437923987
[19]Zhang, J.: Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Commun. partial diff. Eq. 30, 1429-1443 (2005) · Zbl 1081.35109 · doi:10.1080/03605300500299539