The purpose of this study is to construct wave processes that could be identified with non-uniformly moving objects in our world. This is necessary for further development of ideas about the wave nature of matter and elucidation of the causes of Lorentz effects.

Methods. A sequence of active Poincar´e transformations with infinitesimal changes in parameters is applied to the solution of the sine-Gordon equation in the form of a breather, and a limit transition is made in order to obtain a continuous transformation.

Results. Formulas for continuous active Poincar´e transformations were obtained. An equation was derived that describes a wave process in which a selected phase ("peak") moves according to a given law. The breather configuration after the peak reverses
and returns to its original point was found.

Conclusion. The use of continuous active Poincar´e transformations is proposed to describe the setting in motion of wave systems. Using a non-uniformly moving breather, the Lorentzian effects of slowing-down of processes and length contraction are illustrated.