Global existence for the defocusing nonlinear Schr\"odinger equations with limit periodic initial data

We consider the Cauchy problem for the defocusing nonlinear Schr\"odinger 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.

[1]  J. Bourgain.  A remark on normal forms and the “I-method” for periodic NLS , 2004 .

[2]  Terence Tao,et al.  Sharp global well-posedness for KdV and modified KdV on ℝ and , 2003 .

[3]  A. Besicovitch.  Almost Periodic Functions , 1954 .

[4]  J. Ginibre,et al.  On a class of non linear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3 , 1978 .

[5]  H. Takaoka,et al.  Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$ , 2001 .

[6]  Y. Katznelson.  An Introduction to Harmonic Analysis: Interpolation of Linear Operators , 1968 .

[7]  Takayoshi Ogawa,et al.  Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition , 1990 .

[8]  Tadahiro Oh,et al.  On Nonlinear Schrödinger Equations with Almost Periodic Initial Data , 2014, SIAM J. Math. Anal..

[9]  Y. Tsutsumi.  L$^2$-Solutions for Nonlinear Schrodinger Equations and Nonlinear Groups , 1985 .

[10]  KOTARO TSUGAWA.  Local Well-Posedness of the KdV Equation with Quasi-Periodic Initial Data , 2012, SIAM J. Math. Anal..

[11]  Paul H. Rabinowitz,et al.  On a class of nonlinear Schrödinger equations , 1992 .

[12]  David Damanik,et al.  On the Existence and Uniqueness of Global Solutions for the KdV Equation with Quasi-Periodic Initial Data , 2012, 1212.2674.

[13]  Iryna Egorova,et al.  The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense , 1994 .

[14]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[15]  Takayoshi Ogawa,et al.  Blow-up of $H\sp 1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity , 1991 .

[16]  Harald Bohr,et al.  Zur theorie der fast periodischen funktionen , 1925 .

[17]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[18]  T. Tao.  Nonlinear dispersive equations : local and global analysis , 2006 .

[19]  Anne Boutet de Monvel,et al.  On solutions of nonlinear Schrödinger equations with Cantor-type spectrum , 1997 .

[20]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .