A class of exact solutions of Einstein - Maxwell equations which describe the collision and nonlinear interaction of gravitational and electromagnetic soliton waves with strong non-soliton electromagnetic traveling waves of arbitrary amplitudes and profiles propagating in the expanding background space-time is presented. In contrast to intuitive expectations that rather strong traveling waves can destroy the soliton, it occurs that the soliton
survives during its interaction with electromagnetic waves of arbitrary amplitude and profile, but its parameters begin to evolve under the influence of this interaction. If a traveling electromagnetic wave possesses a finite duration, the soliton parameters after interaction take constant values again, but these values in general are different from those before the interaction. These solutions of the Einstein-Maxwell equations demonstrate a series of nonlinear phenomena, such as (a) creation of gravitational waves in the collision of two electromagnetic waves, (b) creation of electromagnetic soliton waves in the collision of a gravitational soliton with traveling electromagnetic waves, (c) scattering of a part of a soliton wave in the direction of propagation of a traveling electromagnetic wave, and (d) quasi-periodic oscillating character of fields in the wave interaction region and multiple mutual transformations of gravitational and electromagnetic waves in this region.