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 [1, 2, 3], has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo-Riesz time-space-fractional nonlinear wave equation [4], in which the 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 is considered, where we explore the sine-Gordon nonlinearity with smooth initial data. For the Riesz and Caputo derivative coefficients α=β=2, we naturally retrieve the exact, analytical form of breather waves expected from the literature. 

Focusing on the Caputo temporal derivative variation within 1<β<2 values for α=2, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of β. 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.