  Quantum walk (QW) provides a versatile tool to study fundamental physics and
also to make a variety of practical applications. We here start with the recent
idea of {\it nonlinear} QW and show that introducing {\it nonlinearity} to QW
can lead to a wealth of remarkable possibilities, e.g., simulating nonlinear
quantum dynamics thus enhancing the applicability of QW above the existing
level for a universal quantum simulator. As an illustration, we show that the
dynamics of a nonlinear Dirac particle can be simulated on an optical nonlinear
QW platform implemented with a measurement-based-feedforward scheme. The
nonlinear evolution induced by the feed-forward introduces a self-coupling
mechanism to (otherwise linear) Dirac particles, which accordingly behave as a
\emph{soliton}. We particularly consider two kinds of nonlinear Dirac
equations, one with a scalar-type self-coupling (Gross-Neveu model) and the
other with a vector-type one (Thirring model), respectively. Using their known
stationary solutions, we confirm that our nonlinear QW framework is capable of
exhibiting characteristic features of a soliton. Furthermore, we show that the
nonlinear QW enables us to observe and control an enhancement and suppression
of the ballistic diffusion.
