In this paper, we continue our study [B. Dodson, A. Soffer, and T. Spencer, J. Stat. Phys. 180, 910 (2020)] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on was proved for real analytic data. Here, we prove global well-posedness for the 1D NLS with initial data lying in Lp for any 2 < p < ∞, provided that the initial data are sufficiently smooth. We do not use the complete integrability of the cubic NLS.
REFERENCES
1.
B.
Dodson
, A.
Soffer
, and T.
Spencer
, “The nonlinear Schrödinger equation on Z and R with bounded initial data: Examples and conjectures
,” J. Stat. Phys.
180
, 910
(2020
).2.
T.
Cazenave
, Semilinear Schrodinger Equations
(American Mathematical Society
, 2003
), Vol. 10.3.
B.
Dodson
, Defocusing Nonlinear Schrödinger Equations
(Cambridge University Press
, 2019
), Vol. 217.4.
T.
Tao
, Nonlinear Dispersive Equations: Local and Global Analysis
(American Mathematical Society
, 2006
), Vol. 106.5.
T.
Oh
, “On nonlinear Schrödinger equations with almost periodic initial data
,” SIAM J. Math. Anal.
47
, 1253
–1270
(2015
).6.
D.
Damanik
and M.
Goldstein
, “On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data
,” J. Am. Math. Soc.
29
, 825
–856
(2016
).7.
A. B.
de Monvel
and I.
Egorova
, “On solutions of nonlinear Schrödinger equations with Cantor-type spectrum
,” J. Anal. Math.
72
, 1
–20
(1997
).8.
T.
Oh
, “Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data
,” Commun. Pure Appl. Anal.
14
, 1563
–1580
(2015
).9.
B.
Harrop-Griffiths
, R.
Killip
, and M.
Visan
, “Sharp well-posedness for the cubic NLS and mKdV in Hs(R)
,” arXiv:2003.05011 (2020
).10.
J.
Bourgain
, “Refinements of Strichartz’s inequality and applications to 2D-NLS with critical nonlinearity
,” Int. Math. Res. Not.
1998
, 253
–283
.11.
J.
Bourgain
, “Invariant measures for the 2D-defocusing nonlinear Schrödinger equation
,” Commun. Math. Phys.
176
, 421
–445
(1996
).12.
A.
Vargas
and L.
Vega
, “Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L2 norm
,” J. Math. Pures Appl.
80
, 1029
–1044
(2001
).13.
Á.
Bényi
, T.
Oh
, and O.
Pocovnicu
, “Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R3
,” Trans. Am. Math. Soc., Ser. B
6
, 114
–160
(2019
).14.
T.
Oh
, O.
Pocovnicu
, and N.
Tzvetkov
, “Probabilistic local well-posedness of the cubic nonlinear wave equation in negative sobolev spaces
,” arXiv:1904.06792 (2019
).15.
N.
Burq
and N.
Tzvetkov
, “Probabilistic well-posedness for the cubic wave equation
,” J. Eur. Math. Soc.
16
, 1
–30
(2013
).16.
B.
Dodson
, “Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space
,” arXiv:2004.09618 (2020
).17.
R.
Hyakuna
, “Local and global well-posedness, and Lp′-decay estimates for 1D nonlinear Schrödinger equations with Cauchy data in Lp
,” J. Funct. Anal.
278
, 108511
(2020
).18.
Y.
Zhou
, “Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space Ws,p for p < 2
,” Trans. Am. Math. Soc.
362
, 4683
–4694
(2010
).19.
R. S.
Strichartz
, “Restrictions of fourier transforms to quadratic surfaces and decay of solutions of wave equations
,” Duke Math. J.
44
, 705
–714
(1977
).20.
J.
Bourgain
, Global Solutions of Nonlinear Schrodinger Equations
(American Mathematical Society
, 1999
), Vol. 46.© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.
