In this paper, we consider effects of impurity diffusion and convection in strained elastic materials with the help of a two-component continual model, that takes into account change in the rigid properties of the material. Two new kinds of solutions, which describe propagation of localized waves, have been found. The first type of solutions describe waves which look like sharply localized peaks (solitary waves). When these type of waves propagate, they change their forms and, as a result, a formation of new peaks is possible. The velocity of the localized waves changes in time and is always less than the sound velocity in the material without impurities. The second kind of solution can be interpreted as shock waves (kinks). The formation mechanism of those waves and their structure are similar to waves of the famous Burgers model; however, their analytical forms are more complicated. They describe jumps in impurity density and deformation.