July  2015, 14(4): 1563-1580. doi: 10.3934/cpaa.2015.14.1563

Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data

1. 

School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom

Received  July 2014 Revised  December 2014 Published  April 2014

We consider the Cauchy problem for the defocusing nonlinear Schrödinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS with limit periodic functions as initial data under some regularity assumption.
Citation: Tadahiro Oh. Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1563-1580. doi: 10.3934/cpaa.2015.14.1563
References:
[1]

A. Besicovitch, Almost Periodic Functions,, Dover Publications, (1955). Google Scholar

[2]

H. Bohr, Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen,, \emph{Acta Math.}, 45 (1925), 29. doi: 10.1007/BF02395468. Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, \emph{Geom. Funct. Anal.}, 3 (1993), 107. doi: 10.1007/BF01896020. Google Scholar

[4]

J. Bourgain, A remark on normal forms and the "$I$-method'' for periodic NLS,, \emph{J. Anal. Math.}, 94 (2004), 125. doi: 10.1007/BF02789044. Google Scholar

[5]

A. Boutet de Monvel and I. Egorova, On solutions of nonlinear Schrödinger equations with Cantor-type spectrum,, \emph{J. Anal. Math.}, 72 (1997), 1. doi: 10.1007/BF02843151. Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar

[7]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). Google Scholar

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar

[9]

C. Corduneanu, Almost Periodic Functions,, With the collaboration of N. Gheorghiu and V. Barbu. Translated from the Romanian by Gitta Bernstein and Eugene Tomer. Interscience Tracts in Pure and Applied Mathematics, (1968). Google Scholar

[10]

D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data,, preprint, (). Google Scholar

[11]

I. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense,, in \emph{Spectral Operator Theory and Related Topics, (1994), 181. Google Scholar

[12]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3,, \emph{Ann. Inst. H. Poincar\'e Sect. A (N.S.)}, 28 (1978), 287. Google Scholar

[13]

Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004). doi: 10.1017/CBO9781139165372. Google Scholar

[14]

T. Ogawa and Y. Tsutsumi, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition,, \emph{Functional-analytic Methods for Partial Differential Equations} (Tokyo, (1989), 236. doi: 10.1007/BFb0084910. Google Scholar

[15]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity,, \emph{Proc. Amer. Math. Soc.}, 111 (1991), 487. doi: 10.2307/2048340. Google Scholar

[16]

T. Oh, On nonlinear Schrödinger equations with almost periodic initial data,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1253. doi: 10.1137/140973384. Google Scholar

[17]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006). Google Scholar

[18]

K. Tsugawa, Local well-posedness of the KdV equation with quasi-periodic initial data,, \emph{SIAM J. Math. Anal.}, 44 (2012), 3412. doi: 10.1137/110849973. Google Scholar

[19]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{ Funkcial. Ekvac.}, 30 (1987), 115. Google Scholar

show all references

References:
[1]

A. Besicovitch, Almost Periodic Functions,, Dover Publications, (1955). Google Scholar

[2]

H. Bohr, Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen,, \emph{Acta Math.}, 45 (1925), 29. doi: 10.1007/BF02395468. Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, \emph{Geom. Funct. Anal.}, 3 (1993), 107. doi: 10.1007/BF01896020. Google Scholar

[4]

J. Bourgain, A remark on normal forms and the "$I$-method'' for periodic NLS,, \emph{J. Anal. Math.}, 94 (2004), 125. doi: 10.1007/BF02789044. Google Scholar

[5]

A. Boutet de Monvel and I. Egorova, On solutions of nonlinear Schrödinger equations with Cantor-type spectrum,, \emph{J. Anal. Math.}, 72 (1997), 1. doi: 10.1007/BF02843151. Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar

[7]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). Google Scholar

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar

[9]

C. Corduneanu, Almost Periodic Functions,, With the collaboration of N. Gheorghiu and V. Barbu. Translated from the Romanian by Gitta Bernstein and Eugene Tomer. Interscience Tracts in Pure and Applied Mathematics, (1968). Google Scholar

[10]

D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data,, preprint, (). Google Scholar

[11]

I. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense,, in \emph{Spectral Operator Theory and Related Topics, (1994), 181. Google Scholar

[12]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3,, \emph{Ann. Inst. H. Poincar\'e Sect. A (N.S.)}, 28 (1978), 287. Google Scholar

[13]

Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004). doi: 10.1017/CBO9781139165372. Google Scholar

[14]

T. Ogawa and Y. Tsutsumi, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition,, \emph{Functional-analytic Methods for Partial Differential Equations} (Tokyo, (1989), 236. doi: 10.1007/BFb0084910. Google Scholar

[15]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity,, \emph{Proc. Amer. Math. Soc.}, 111 (1991), 487. doi: 10.2307/2048340. Google Scholar

[16]

T. Oh, On nonlinear Schrödinger equations with almost periodic initial data,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1253. doi: 10.1137/140973384. Google Scholar

[17]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006). Google Scholar

[18]

K. Tsugawa, Local well-posedness of the KdV equation with quasi-periodic initial data,, \emph{SIAM J. Math. Anal.}, 44 (2012), 3412. doi: 10.1137/110849973. Google Scholar

[19]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{ Funkcial. Ekvac.}, 30 (1987), 115. Google Scholar

[1]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37

[2]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093

[3]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261

[4]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010

[5]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023

[6]

Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815

[7]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905

[8]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[9]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389

[10]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[11]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393

[12]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15

[13]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803

[14]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41

[15]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75

[16]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605

[17]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[18]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100

[19]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

[20]

Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]