  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.
