For non-integrable NLS on R, the paper proves existence of space-time quasi-periodic, infinite-energy, smooth solutions with two frequencies for a large Cantor set of parameters.
The nonlinear Schrodinger equation on Z and R with bounded initial data: examples and conjectures
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abstract
We study the nonlinear Schr\"odinger 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.
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Infinite energy quasi-periodic solutions to nonlinear Schr\"odinger equations on $\mathbb R$
For non-integrable NLS on R, the paper proves existence of space-time quasi-periodic, infinite-energy, smooth solutions with two frequencies for a large Cantor set of parameters.