Dynamics of solitons for nonlinear quantum walks

TL;DR

This paper investigates the behavior of nonlinear quantum walks, revealing decay patterns in weak nonlinearity and soliton formations in strong nonlinearity, including soliton collisions and explicit solutions.

Contribution

It provides new insights into the nonlinear dynamics of quantum walks, especially the emergence and interaction of solitons, extending previous analytical results.

Findings

Linear decay observed for wider nonlinearity range

Soliton solutions appear in strong nonlinear regimes

Collision dynamics between solitons studied

Abstract

We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically \cite{MSSSS18DCDS, MSSSS18QIP}. 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 \cite{MSSSS18DCDS}. 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.