Reported in this paper are results concerning the Cauchy problem and the dynamics for a cubic nonlinear Schrödinger system arising in nonlinear optics. A sharp criterion is given concerned with the dichotomy global existence versus finite time blow-up. When a radial solution blows up in finite time, we prove the concentration in the critical Lebesgue space. Sufficient condition for the scattering and the construction of the wave operator in the energy space is also provided.
REFERENCES
1.
2.
P.
Bégout
, “Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation
,” Adv. Math. Sci. Appl.
12
, 817
–827
(2002
).3.
J.
Bergh
and J.
Löfsröm
, Interpolation Spaces: An introduction
(Springer-Verlag
, New York
, 1976
).4.
B.
Cassano
and M.
Tarulli
, “H1scattering for systems of N-defocusing weakly coupled NLS equations in low space dimensions,” e-print arXiv:1409.8416.5.
T.
Cazenave
, Semilinear Schrödinger Equations
, Courant Lectures Notes Vol. 10
(American Mathematical Society
, Providence
, 2003
).6.
J.
Chen
and B.
Guo
, “Blow-up profile to the solutions of two-coupled Schrödinger equations
,” J. Math. Phys.
50
, 023505
(2009
).7.
F. M.
Christ
and M. I.
Weinstein
, “Dispersion and small amplitude solutions of the generalized Korteweg-de Vries equation
,” J. Funct. Anal.
100
, 87
–109
(1991
).8.
L.
Fanelli
and E.
Montefusco
, “On the blow-up threshold for weakly coupled nonlinear Schrödinger equations
,” J. Phys. A: Math. Theor.
40
, 14139
–14150
(2007
).9.
A.
Friedman
, Partial Differential Equations
(Holt, Rinehart and Winston
, New York
, 1969
).10.
J.
Holmer
and S.
Roudenko
, “A sharp condition for scattering of the radial 3D nonlinear Schrödinger equation
,” Commun. Math. Phys.
282
, 435
–467
(2008
).11.
J.
Holmer
and S.
Roudenko
, “On blow-up solutions to the 3D cubic nonlinear Schrödinger equation
,” Appl. Math. Res. Express.
2007, 1–31 (2007).12.
O.
Kavian
, “A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations
,” Trans. Am. Math. Soc.
299
, 193
–203
(1987
).13.
C. E.
Kenig
and F.
Merle
, “Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
,” Invent. Math.
166
, 645
–675
(2006
).14.
C. E.
Kenig
, G.
Ponce
, and L.
Vega
, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle
,” Commun. Pure Appl. Math.
46
, 527
–620
(1993
).15.
R.
Killip
and M.
Visan
, “The radial defocusing energy-supercritical nonlinear wave equation in all sapce dimensions
,” Proc. Am. Math. Soc.
139
, 1805
–1817
(2011
).16.
X.
Li
, Y.
Wu
, and S.
Lai
, “A sharp threshold of blow-up for coupled nonlinear Schrödinger equations
,” J. Phys. A: Math. Theor.
43
, 165205
(2010
).17.
Z.
Lü
and Z.
Liu
, “L2-concentration of blow-up solutions for two-coupled nonlinear Schrödinger equations
,” J. Math. Anal. Appl.
380
, 531
–539
(2011
).18.
L.
Ma
and L.
Zhao
, “Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system
,” J. Math. Phys.
49
, 062103
(2008
).19.
L. A.
Maia
, E.
Montefusco
, and B.
Pellacci
, “Positive solutions for a weakly coupled nonlinear Schrödinger system
,” J. Differ. Equations
229
, 743
–767
(2006
).20.
F.
Merle
and Y.
Tsutsumi
, “L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity
,” J. Differ. Equations
84
, 205
–214
(1990
).21.
B. V. Sz.
Nagy
, “Über Integralgleichungen zwischen einer Funktion und ihrer Ableitung
,” Acta Univ. Szeged. Sect. Sci. Math.
10
, 64
–74
(1941
).22.
N. V.
Nguyen
, R.
Tian
, B.
Deconinck
, and N.
Sheils
, “Global existence for a coupled system of Schrödinger equations with power-type nonlinearities
,” J. Math. Phys.
54
, 011503
(2013
).23.
T.
Ogawa
and Y.
Tsutsumi
, “Blow-up of H1 solutions for the nonlinear Schrödinger equation
,” J. Differ. Equations
92
, 317
–330
(1991
).24.
V.
Prytula
, V.
Vekslerchik
, and V. M.
Pérez-García
, “Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications
,” Physica D
238
, 1462
–1467
(2009
).25.
X.
Song
, “Sharp thresholds of global existence and blowup for a system of Schrödinger equations with combined power-type nonlinearities
,” J. Math. Phys.
51
, 033509
(2010
).26.
W.
Strauss
, “Nonlinear scattering theory at low energy
,” J. Funct. Anal.
41
, 110
–133
(1981
).27.
W.
Strauss
, Nonlinear Wave Equations
, CBMS Regional Conference Series in Mathematics Vol. 73
(American Mathematical Society
, Providence
, 1989
).28.
W.
Strauss
, “Existence of solitary waves in higher dimensions
,” Commun. Math. Phys.
55
, 149
–162
(1977
).29.
M. I.
Weinstein
, “Nonlinear Schrödinger equations and sharp interpolation estimates
,” Commun. Math. Phys.
87
, 567
–576
(1983
).© 2015 AIP Publishing LLC.
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