The Nonlinear Schrödinger 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.
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- May 2020
- DOI:
- 10.1007/s10955-020-02552-w
- arXiv:
- arXiv:2003.08849
- Bibcode:
- 2020JSP...180..910D
- Keywords:
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- Propagation speed;
- Well-posedness;
- Local conservation laws;
- Newton iteration;
- Mathematics - Analysis of PDEs;
- Mathematical Physics;
- 35Q55
- E-Print:
- 22 pages