  We study the dynamics of a nonlinear two-level crossing model with a cubic
modification of the linear Landau-Zener diabatic energies. The solutions are
expressed in terms of the bi-confluent Heun functions --- the generalization of
the confluent hypergeometric functions. We express the finial transition
probability as a convergent series of the parameters of the nonlinear laser
detuning, and derive analytical approximations for the state populations in
terms of parabolic cylinder and Whittaker functions. Tractable closed-form
expressions are derived for a large part of the parameter space. We also
provide simple method to determine the transition point which connects local
solutions in different physical limits. The validity of the analytical
approximations is shown by comparison with numerical results of simulations.
