# American Institute of Mathematical Sciences

July  2018, 38(7): 3687-3703. doi: 10.3934/dcds.2018159

## Scattering and inverse scattering for nonlinear quantum walks

 1 Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan 2 Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan 3 Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Nagano 380-8553, Japan 4 College of Science, Ibaraki University, 2-1-1 Bunkyo, Mito 310-8512, Japan

Received  December 2017 Published  April 2018

Fund Project: M.M. was supported by the JSPS KAKENHI Grant Numbers JP15K17568, JP17H02851 and JP17H02853. H.S. was supported by JSPS KAKENHI Grant Number JP17K05311. E.S. acknowledges financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant No. 16K17637, No. 16K03939). A. S. was supported by JSPS KAKENHI Grant Number JP26800054. K.S acknowledges JSPS the Grant-in-Aid for Scientific Research (C) 26400156.

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.

Citation: Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
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