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The nonlinear Schrödinger equation on Z and R with bounded initial data: examples and conjectures. (English) Zbl 1446.35181

Summary: 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B65 Smoothness and regularity of solutions to PDEs
35C08 Soliton solutions
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
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References:

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