  I classify multiphoton absorption into separable, linked, and simultaneous
processes. The first and second types can be distinguished when the
rate-equation approximation is valid whereas the third type refers to the case
when the full description of multiphoton absorption is essential. For this
purpose, rate equations are solved analytically without decay processes which
shows that even if many photons are absorbed the interaction with the light
field is linear and one has the case of separable multiphoton absorption. Next
a short-pulse approximation is investigated in which I first solve the rate
equations without decay processes and then solve only rate equations for the
ensuing decay. Finally, the full rate equations are examined and a successive
approximation of the underlying Volterra integral equation of the second kind
is derived leading to linked multiphoton absorption by the involved decay
widths. The three methods are applied to a nitrogen atom in intense and
ultrafast x rays from free-electron lasers (FELs). The linearity theorem barely
approximates the results in the presence of decay processes which is also not
satisfactorily corrected for by the short-pulse approximation. The successive
approximation gives excellent agreement with the numerically-exact solution of
the rate equations.
