  A nonlinear Landau-Zener model was proposed recently to describe, among a
number of applications, the nonadiabatic transition of a Bose-Einstein
condensate between Bloch bands. Numerical analysis revealed a striking
phenomenon that tunneling occurs even in the adiabatic limit as the nonlinear
parameter $C$ is above a critical value equal to the gap $V$ of avoided
crossing of the two levels. In this paper, we present analytical results that
give quantitative account of the breakdown of adiabaticity by mapping this
quantum nonlinear model into a classical Josephson Hamiltonian. In the critical
region, we find a power-law scaling of the nonadiabatic transition probability
as a function of $C/V-1$ and $\alpha $, the crossing rate of the energy levels.
In the subcritical regime, the transition probability still follows an
exponential law but with the exponent changed by the nonlinear effect. For
$C/V>>1$, we find a near unit probability for the transition between the
adiabatic levels for all values of the crossing rate.
