We present a comprehensive analysis of the Landau-Zener tunnelling of a
nonlinear three-level system in a linearly sweeping external field. We find the
presence of nonzero tunnelling probability in the adiabatic limit (i.e., very
slowly sweeping field) even for the situation that the nonlinear term is very
small and the energy levels keep the same topological structure as that of
linear case. In particular, the tunnelling is irregular with showing an
unresolved sensitivity on the sweeping rate. For the case of fast-sweeping
fields, we derive an analytic expression for the tunnelling probability with
stationary phase approximation and show that the nonlinearity can dramatically
influence the tunnelling probability when the nonlinear "internal field"
resonate with the external field. We also discuss the asymmetry of the
tunnelling probability induced by the nonlinearity. Physics behind the above
phenomena is revealed and possible application of our model to triple-well
trapped Bose-Einstein condensate is discussed.Comment: 8 pages, 8 figure