The nonlinear Schrodinger equation on Z and R with bounded initial data: examples and conjectures

Abstract

We study the nonlinear Schrödinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.

arXiv comment22 pages

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Cited by in corpus (5)

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• Spatially quasi-periodic water waves of finite depth
• The Quasi-Periodic Cauchy Problem for the Generalized Benjamin-Bona-Mahony Equation on the Real Line
• Infinite energy quasi-periodic solutions to nonlinear Schrödinger equations on $\mathbb{R}$