© 2024 The Author(s)In honor of the great Russian mathematician A. N. Kolmogorov, we would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time evolution for integer derivatives has a time-honored conservative form. This observation, although known to the general mathematical community, (Achar et al., 2001; Stanislavsky, 2004; Diethelm and Ford, 2010; Chung and Jung, 2014; Olivar-Romero and Rosas-Ortiz, 2017; Baleanu et al., 2020) has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo–Riesz time–space-fractional nonlinear wave equation of Macias Diaz (2022), in which two of the present authors introduced an energy-type functional and proposed a finite-difference scheme to approximate the solutions of the continuous model. The relevant Klein–Gordon equation considered here has the form: [Formula presented] where we explore the sine-Gordon nonlinearity F(?)=1?cos(?) with smooth initial data. For ?=?=2, we naturally retrieve the exact, analytical form of breather waves expected from the literature. Focusing on the Caputo temporal derivative variation within 1<?<2 values for ?=2, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of ?. We compare such findings to single degree-of-freedom linear and nonlinear oscillators in the presence of Caputo temporal derivatives and also consider anti-damping mechanisms to counter the relevant effect. These findings also motivate some interesting directions for further study, e.g., regarding the consideration of topological solitary waves, such as kinks/antikinks and their dynamical evolution in this model.