Abstract
We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703; Maeda et al 2018 Quantum Inf. Process. 17 215). It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703). In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.
Highlights
Quantum walks (QWs), which are quantum analog of classical random walks [2, 9, 12, 21], are attracting much interest because of its connection to various regimes in mathematics, physics and applications such as quantum algorithms [3, 6, 23] and topological insulators [4, 5, 8, 11, 14, 15, 16]
Nonlinear quantum walks (NLQWs), which are nonlinear versions of QWs, have been recently proposed by several authors [10, 13, 22] and in particular related to some nonlinear differential equations such as nonlinear Dirac equations [13]
We have studied the dynamics of solutions of NLQWs in a strong nonlinear regime
Summary
Quantum walks (QWs), which are quantum analog of classical random walks [2, 9, 12, 21], are attracting much interest because of its connection to various regimes in mathematics, physics and applications such as quantum algorithms [3, 6, 23] and topological insulators [4, 5, 8, 11, 14, 15, 16]. In [18], we have initiated an analytical study of NLQWs using the methods developed for the study of nonlinear dispersive equations. It was shown in [18] that the scattering phenomena for NLQWs in a weak nonlinear regime, that is, the behavior of solutions of NLQWs can be approximated by that of corresponding to linear quantum walks.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
