2024/06/13 by Tassos Bountis, Bountis, Tassos, Julia Cantisán +7
Mathematics · #Differential Equations and Numerical Methods #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Numerical methods for differential equations #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2406.08912
openalex publication_date 2024/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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: \frac ∂ βϕ(x , t) ∂ t β - Δαϕ(x , t) + F ^′ (ϕ(x , t)) = 0, ∀ (x , t) ∈ (-∞,∞) where we explore the sine-Gordon nonlinearity F(ϕ)=1-cos(ϕ) with smooth initial data. For 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.