The Dissipative Effect of Caputo--Time-Fractional Derivatives and its Implications for the Solutions of Nonlinear Wave Equations

Abstract

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, has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo-Riesz time-space-fractional nonlinear wave equation, 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: \begin{equation} \frac {\partial ^\beta \phi (x , t)} {\partial t ^\beta} - \Delta ^\alpha \phi (x , t) + F ^\prime (\phi (x , t)) = 0, \quad \forall (x , t) \in (-\infty,\infty) \end{equation} where we explore the sine-Gordon nonlinearity $F(\phi)=1-\cos(\phi)$ with smooth initial data. For $\alpha=\beta=2$, we naturally retrieve the exact, analytical form of breather waves expected from the literature. Focusing on the Caputo temporal derivative variation within $1< \beta < 2$ values for $\alpha=2$, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of $\beta$. 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.