Abstract
We consider the Cauchy problem for the defocusing nonlinear Schrodinger 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.
Highlights
We consider the Cauchy problem for the defocusing nonlinear Schrodinger equations (NLS) on R: i∂tu + ∂x2u = |u|2ku, u|t=0 = f, (t, x) ∈ R × R, (1.1)for k ∈ N
Our main interest in this paper is to study global-in-time behavior of solutions to the Cauchy problem (1.1) with a particular subclass of almost periodic functions as initial data
We introduce limit periodic functions and state our main result (Theorem 1.9) in Subsection 1.2
Summary
To the best of the author’s knowledge, Theorem 1.9 is the first global existence result for NLS (1.1), k ≥ 2, with limit periodic functions as initial data. We prove Theorem 1.9 by combining global well-posedness of the defocusing NLS in the periodic setting and scaling invariance. The space Sp, p ≥ 1, of Stepanov’s generalized almost periodic functions is precisely the closure of the trigonometric polynomials under the Sp-metric dSp defined by dSp(f, g) := supy+1 |f (x) − g(x)|pdx p (1.19). Note that this metric is induced by the Ss,p-norm with s = 0. We refer to f (L) as the periodization of f (with period L)
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