We consider x-ray-hydrogen atom scattering in the presence of a monoch
romatic linearly polarized laser field. The S matrix of this process i
s presented and an expression for the differential cross section (DCS)
is derived. We show that the time-dependent Wentzel-Kramers-Brillouin
approximation can be applied to the present problem. The presented nu
merical results for the DCS as a function of the number n of photons e
xchanged with the laser field show a characteristic behavior. The numb
er n can only be even. For n = 0, +/-2 we have pronounced maxima in th
e DCS, followed by sharp minima at n=+/-4. After that, Eve have a plat
eau that is different for negative and positive values of n. The plate
au for negative Values of n is much more extended than for the positiv
e ones. The structure of the plateau and the positions of the minima a
nd maxima of the DCS do not depend on the laser field intensity, while
the height of the plateaus strongly depends on it. We also analyze th
e dependence of the matrix elements of the x-ray spectra on the incide
nt x-ray photon energies by using bath the saddle-point method and the
numerical evaluation, and we show that the shape and position of the
plateau are determined by the simple relation n (h) over bar omega=I-0
-(h) over bar omega(K). This condition connects the number of absorbed
or emitted laser field photons n, the laser field photon energy Aw, t
he atomic ionization potential I-0, and the energy of the incident x-r
ay photon (h) over bar omega K.