In this work, we study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations with quadratic nonlinearities, which appears modeling problems in nonlinear optics. We obtain the local well-posedness for data in Sobolev spaces with low regularity. To obtain the local theory, we prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, we establish the global well-posedness in , for some negatives indexes s. The proof of our global result uses the I-method introduced by Colliander et al.
REFERENCES
1.
Angulo
, J.
and Linares
, F.
, “Periodic pulses of coupled nonlinear Schrödinger equations in optics
,” Indiana Univ. Math. J.
56
(2
), 847
–878
(2007
).2.
Bourgain
, J.
, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
,” Geom. Funct. Anal.
3
(3
), 209
–262
(1993
).3.
Colin
, M.
, Colin
, T.
, and Ohta
, M.
, “Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction
,” Ann. Inst. Henri Poincare (C) Non Linear Anal.
26
(6
), 2211
–2226
(2009
).4.
Colliander
, J.
, Keel
, M.
, Staffilani
, G.
, Takaoka
, H.
, and Tao
, T.
, “Global well-posedness for KdV in Sobolev spaces of negative index
,” Electron. J. Differ. Equations
2001
(26
), 1
–7
.5.
Colliander
, J.
, Keel
, M.
, Staffilani
, G.
, Takaoka
, H.
, and Tao
, T.
, “Global well-posedness for Schrödinger equations with derivative
,” SIAM J. Math. Anal.
33
(3
), 649
–669
(2001
).6.
Colliander
, J.
, Keel
, M.
, Staffilani
, G.
, Takaoka
, H.
, and Tao
, T.
, “Multilinear estimates for periodic KdV equations, and applications
,” J. Funct. Anal.
211
(1
), 173
–218
(2004
).7.
Corcho
, A. J.
and Linares
, F.
, “Well-posedness for the Schrödinger-Korteweg-de Vries system
,” Trans. Am. Math. Soc.
359
(9
), 4089
–4106
(2007
).8.
Corcho
, A. J.
and Matheus
, C.
, “Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system
,” Differ. Integr. Equations
22
(3-4
), 357
–391
(2009
).9.
DeSalvo
, R.
, Vanherzeele
, H.
, Hagan
, D.
, Sheik-Bahae
, M.
, Stegeman
, G.
, and Van Stryland
, E.
, “Self-focusing and self-defocusing by cascaded second-order effects in KTP
,” Opt. Lett.
17
(1
), 28
–30
(1992
).10.
Ginibre
, J.
, Tsutsumi
, Y.
, and Velo
, G.
, “On the Cauchy problem for the Zakharov system
,” J. Funct. Anal.
151
(2
), 384
–436
(1997
).11.
Hayashi
, N.
, Ozawa
, T.
, and Tanaka
, K.
, “On a system of nonlinear Schrödinger equations with quadratic interaction
,” Ann. Inst. Henri Poincare (C) Non Linear Anal.
30
(4
), 661
–690
(2013
).12.
Karamzin
, Y. N.
and Sukhorukov
, A.
, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: Mutual focusing of beams and limitation on the efficiency of optical frequency converters
,” JETP Lett.
20
(11
), 339
–343
(1974
).13.
Kenig
, C.
, Ponce
, G.
, and Vega
, L.
, “Quadratic forms for the 1-D semilinear Schrödinger equation
,” Trans. Am. Math. Soc.
348
(8
), 3323
–3353
(1996
).14.
Li
, C.
and Hayashi
, N.
, “Recent progress on nonlinear Schrödinger systems with quadratic interactions
,” Sci. World J.
2014
, 1
.15.
Menyuk
, C.
, Schiek
, R.
, and Torner
, L.
, “Solitary waves due to χ (2): χ (2) cascading
,” J. Opt. Soc. Am. B
11
(12
), 2434
–2443
(1994
).16.
Pecher
, H.
, “The Cauchy problem for a Schrödinger-Korteweg-de Vries system with rough data
,” Differ. Integr. Equations
18
(10
), 1147
–1174
(2005
).17.
Yew
, A.
, “Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations
,” Indiana Univ. Math. J.
49
(3
), 1079
–1124
(2000
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
