The (nonintegrable) nonlinear Schrödinger equation with saturable nonlinearity |phi|^2/(1+|phi|^2), which we call the saturable Schrödinger equation, or sat-NLS, supports solitary waves which we call saturons. Numerical results show that collisions of these waves can preserve their identities and at the same time exchange information, but such collisions can also generate radiation in amounts dependent on the velocities and phases of the colliding saturons. We compare the sat-NLS with the more commonly studied integrable 3-NLS equation, which describes solitons in optical fibers and supports non-radiating soliton collisions that transfer very limited information. We discuss possible physical realization of the sat-NLS, and potential application of saturon collisions to a new kind of computer based on direct nonlinear wave interaction in a uniform medium. This is possible using either temporal or spatial saturons.