  We derive five classes of quantum time-dependent two-state models solvable in
terms of the double confluent Heun functions, five other classes solvable in
terms of the biconfluent Heun functions, and a class solvable in terms of the
triconfluent Heun functions. These classes generalize all the known families of
two- or three-parametric models solvable in terms of the confluent
hypergeometric functions to more general four-parametric classes involving
three-parametric detuning modulation functions. The particular models derived
describe different non-linear (parabolic, cubic, sinh, cosh, etc.)
level-sweeping or level-glancing processes, double- or triple-level-crossing
processes, as well as periodically repeated resonance-glancing or
resonance-crossing processes. We show that more classes can be derived using
the equations obeyed by certain functions involving the derivatives of the
confluent Heun functions. We present an example of such a class for each of the
three discussed confluent Heun equations.
