  For quantum computers to become useful tools to physicists, engineers and
computational scientists, quantum algorithms for solving nonlinear differential
equations need to be developed. Despite recent advances, the quest for a solver
that can integrate nonlinear dynamical systems with a quantum advantage, whilst
being realisable on available (or near-term) quantum hardware, is an open
challenge. In this paper, we propose to transform a nonlinear dynamical system
into a linear system, which we integrate with quantum algorithms. Key to the
method is the Fokker-Planck equation, which is a non-normal partial
differential equation. Three integration strategies are proposed: (i)
Forward-Euler stepping by unitary block encoding; (ii) Schroedingerisation, and
(iii) Forward-Euler stepping by linear addition of unitaries. We emulate the
integration of prototypical nonlinear systems with the proposed quantum
solvers, and compare the output with the benchmark solutions of classical
integrators. We find that classical and quantum outputs are in good agreement.
This paper opens opportunities for solving nonlinear differential equations
with quantum algorithms.
