Scattering and inverse scattering for nonlinear quantum walks

Masaya Maeda; Hironobu Sasaki; Etsuo Segawa; Akito Suzuki; Kanako Suzuki

doi:10.3934/dcds.2018159

Article Contents

2018, Volume 38, Issue 7: 3687-3703. Doi: 10.3934/dcds.2018159

This issue Previous Article Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise Next Article Corrigendum to: On specification and measure expansiveness

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: July 2018

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 81Q05; Secondary: 35Q55.

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