The Dissipative Effect of Caputo--Time-Fractional Derivatives and its Implications for the Solutions of Nonlinear Wave Equations
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917692620931072 |
|---|---|
| author | Bountis, Tassos Cantisán, Julia Cuevas-Maraver, Jesús Macías-Díaz, J. E. Kevrekidis, Panayotis G. |
| author_facet | Bountis, Tassos Cantisán, Julia Cuevas-Maraver, Jesús Macías-Díaz, J. E. Kevrekidis, Panayotis G. |
| contents | 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 ^βϕ(x , t)} {\partial t ^β} - Δ^αϕ(x , t) + F ^\prime (ϕ(x , t)) = 0, \quad \forall (x , t) \in (-\infty,\infty) \end{equation} 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08912 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dissipative Effect of Caputo--Time-Fractional Derivatives and its Implications for the Solutions of Nonlinear Wave Equations Bountis, Tassos Cantisán, Julia Cuevas-Maraver, Jesús Macías-Díaz, J. E. Kevrekidis, Panayotis G. Pattern Formation and Solitons Mathematical Physics 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 ^βϕ(x , t)} {\partial t ^β} - Δ^αϕ(x , t) + F ^\prime (ϕ(x , t)) = 0, \quad \forall (x , t) \in (-\infty,\infty) \end{equation} 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. |
| title | The Dissipative Effect of Caputo--Time-Fractional Derivatives and its Implications for the Solutions of Nonlinear Wave Equations |
| topic | Pattern Formation and Solitons Mathematical Physics |
| url | https://arxiv.org/abs/2406.08912 |