  Non-degenerate two-photon absorption (TPA) is investigated in a nanophotonic
silicon waveguide in a configuration such that the dispersion of the nonlinear
absorption and refraction cannot be neglected. It is shown that a signal wave
can strongly be absorbed by cross-TPA by interaction with a low energy pump
pulse (1.2\,pJ), close to the half-bandgap and experiencing low nonlinear
absorption. The experiments are very well reproduced by numerical simulations
of two-coupled generalized nonlinear Schr\"{o}dinger equations (GNLSE),
validating the usual approximation made to compute the cross nonlinear
coefficients in indirect-gap semiconductors. We show that the nonlinear
dynamics can be well described by a single GNLSE despite the wavelength
separation between the pump and the signal waves. We also demonstrate that in
silicon wire waveguides and contrary to optical fibers, the dispersion of the
nonlinear absorption is much larger than the dispersion of the Kerr effect.
This could have an impact in the design of all-optical functions based on
cross-TPA, as well as on the study of supercontinuum and frequency comb
generation in integrated semiconductor-on-insulator platforms.
