On weak interaction between a ground state and a trapping potential
Scipio Cuccagna Masaya Maeda
Discrete & Continuous Dynamical Systems - A 2015, 35(8): 3343-3376 doi: 10.3934/dcds.2015.35.3343
We continue our study initiated in [4] of the interaction of a ground state with a potential considering here a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.
keywords: ground states Nonlinear Schrödinger equation asymptotic stability.
Scattering and inverse scattering for nonlinear quantum walks
Masaya Maeda Hironobu Sasaki Etsuo Segawa Akito Suzuki Kanako Suzuki
Discrete & Continuous Dynamical Systems - A 2018, 38(7): 3687-3703 doi: 10.3934/dcds.2018159

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QWs.

keywords: Quantum walks scattering theory dispersive estimates Strichartz estimates nonlinear scattering

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