The dissipative effect of Caputo–time-fractional derivatives and its implications for the solutions of nonlinear wave equations

Contents lists available at ScienceDirect Partial Differential Equations in Applied Mathematics journal homepage: www.elsevier.com/locate/padiff The dissipative effect of Caputo–time-fractional derivatives and its implications for the solutions of nonlinear wave equations Tassos Bountisa,b,∗, Julia Cantisánc, Jesús Cuevas-Maraver d,e, J.E. Macías-Díaz f,g, Panayotis G. Kevrekidis h aDepartment of Mathematics, University of Patras, 26500 Patras, Greece bCenter for Integrable Systems, P.G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia cNonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, C/ Tulipán s/n, 28933 Móstoles, Madrid, Spain dGrupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/ Virgen de África, 7, 41011 Sevilla, Spain eInstituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain fDepartment of Mathematics and Didactics of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia gDepartamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Ags. 20131, Mexico hDepartment of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-4515, USA ARTICLE INFO MSC: 65M06 65M12 Keywords: Caputo–Riesz time–space-fractional system Generalized nonlinear wave equation Artificial dissipation due to fractional Caputo time-derivatives Breathers Kinks 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, (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: 𝜕𝛽𝜙(𝑥, 𝑡) 𝜕𝑡𝛽−𝛥𝛼𝜙(𝑥, 𝑡) + 𝐹′(𝜙(𝑥, 𝑡)) = 0,∀(𝑥, 𝑡) ∈ (−∞,∞) (0.1) 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 < 2values 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. 1. Introduction The exploration of mathematical models involving fractional derivatives has been progressively expanding and forming a fruitful area of research leading to a very substantial volume of publications in many scientific disciplines. To cite a few among the countless relevant examples, in the epidemiological sciences, relevant models have been ∗Corresponding author at: Department of Mathematics, University of Patras, 26500 Patras, Greece. E-mail addresses: [email protected] (T. Bountis), [email protected] (J. Cantisán), [email protected] (J. Cuevas-Maraver), [email protected], [email protected] (J.E. Macías-Díaz), [email protected] (P.G. Kevrekidis). employed to estimate the spreading of computer viruses,1,2the propagation of measles in human populations,3as well as the transmission dynamics of varicella zoster virus.4Among numerous other areas, such applications arise also in biology (e.g., the investigation of threespecies systems Ref. 5), economics (e.g., the mathematical modeling of Chinese economic growth Ref. 6) and nonlinear optics.7The relevant progress has been summarized by now in a number of reviews and https://doi.org/10.1016/j.padiff.2024.100807 Received 12 June 2024; Accepted 6 July 2024 Partial Dierential Equations in Applied Mathematics 11 (2024) 100807 Available online 14 July 2024 2666-8181/© 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/bync-nd/4.0/ ). T. Bountis et al. books,8,9including more specialized volumes, such as one on nonlinear dispersive waves (see, e.g., Ref. 10), which is the topic of the present work. Within the theory of fractional calculus itself, a wide range of fractional operators has been proposed,11,12 leading to the emergence of corresponding analytical results13,14 in the mathematical literature. However, a central question continues to focus on the significance and applicability of such mathematical objects in real-life physical, chemical or/and biological applications, a theme that is still very much under investigation. So far, one can argue that Riesz fractional operators have been adequately justified for their use in physical problems15 and are commonly used to replace ordinary spatial derivatives. They have been proposed as a valid approach to the continuum limit of systems of particles with long-range interactions16 (see also relevant chapters of Ref. 10, such as the one on the long-range variant of the sine-Gordon model we will consider below). Additional applications include, but are not limited to, the study of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potentials17 and the phenomenon of supratransmission.18,19 In addition to the Riesz fractional operator, the one due to Caputo seems to be of great interest to several areas of science and engineering.20–22 In particular, these operators have appeared in the extension of well-known parabolic or hyperbolic equations of mathematical physics, including fractional forms of the sine-Gordon equation,23,24 the nonlinear Klein–Gordon equation,25 the nonlinear Schrödinger equation,10,26 the fractional KdV equation10,27 and the Klein–Gordon– Zakharov system.28 It is worth mentioning that all of these models obey conservation laws which resemble a form of mass or energy conservation of the equivalent systems in the integer-order case.29 It is also worth noting that, from a numerical perspective, a diverse array of methods have employed different tools, including finite differences,30 finite elements,31 spectral techniques,32 among other approaches.33–36 In a recent work,37 two of the present authors considered a fractional extension of the nonlinear Klein–Gordon equation, with Caputo temporal derivatives and Riesz spatial derivatives and showed that this model satisfies a conservation law in differential form. A discretization was subsequently introduced that satisfies the same law in the discrete domain and which leads to a finite-difference model that employs the 𝐿1scheme to approximate the Caputo temporal derivatives,38 and fractional-order centered differences to estimate the Riesz fractional derivatives in space.39 Subsequently, this nonlinear Klein–Gordon equation was solved numerically as a Caputo–Riesz time–space-fractional initial–boundary-value problem with homogeneous Neumann boundary conditions in the form 𝜕𝛽𝜙(𝑥, 𝑡) 𝜕𝑡𝛽−𝛥𝛼𝜙(𝑥, 𝑡) + 𝐹′(𝜙(𝑥, 𝑡)) = 0,∀(𝑥, 𝑡) ∈ 𝛺𝑇, such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜙(𝑥, 0) = 𝜓(𝑥),∀𝑥∈𝛺, 𝜕𝜙(𝑥, 0) 𝜕𝑡 =𝜒(𝑥),∀𝑥∈𝛺, 𝜙(𝑥, 𝑡)=0,∀(𝑥, 𝑡) ∈ 𝜕𝛺 × [0, 𝑇 ]. (1.1) with 𝐹(𝜙) = 1 − cos(𝑢)and various values for the 𝛼and 𝛽orders of the Caputo and Riesz derivatives respectively, using as initial condition the exact breather solution of the 𝛼=𝛽= 2 equation40 for frequency values 0<𝜔<1: 𝜙br (𝑥, 𝑡) = 4 arctan (√1 − 𝜔2cos(𝜔𝑡) 𝜔cosh(√1 − 𝜔2𝑥)),∀(𝑥, 𝑡) ∈ R× [0,∞).(1.2) When the orders of the fractional Caputo and Riesz derivatives are of the classical integer type 𝛼= 2,𝛽= 2, one naturally gets the expected form of the oscillating breather shown in Fig. 1 below. The scope of these simulations was to examine the potential persistence of the relevant waveform in the context of the fractional model. Our purpose in the present numerical study is to offer further systematic insight on the breather dynamics in the fractional (in time, via a Caputo derivative) sine-Gordon model. More specifically, we seek to analyze the dependence on the fractional temporal power 𝛽and how this leads to a remarkable decay of the breather. We compare our findings to prototypical lower dimensional fractional models, such as the one degree-of-freedom fractional pendulum or/and the fractional harmonic oscillator, while at the same time developing suitable diagnostics including the evolution of the breather center, its transition time from elastic to viscous behavior, as well as the potential role of a counter-acting local (dashpot) ‘‘pumping’’ and how that can affect the dynamics. In what follows, we present in Section 2, the model setup, while in Sections 3and 4we expand on our numerical results. Finally, in Section 5, we summarize our findings, present our conclusions and discuss some potential directions for future study. 2. Model setup As discussed above, our main aim herein is to develop one of the main features of the model of Ref. 37, by considering fractionality solely in the time derivative, via a Caputo fractional derivative. In light of that, we will consider the dynamics of breathers in the timefractional sine-Gordon equation with a ‘‘classical’’ (spatial) dispersion, i.e., for 𝛼= 2. In other words, our dynamical equation reads: 𝜕𝛽𝜙(𝑥, 𝑡) 𝜕𝑡𝛽−𝜕2𝜙(𝑥, 𝑡) 𝜕𝑥2+ sin(𝜙(𝑥, 𝑡)) = 0 (2.1) with 𝜕𝛽∕𝜕𝑡𝛽representing the Caputo time-derivative. Depending on the value of 𝛽, several regimes can be observed, even for an uncoupled (one degree-of-freedom) harmonic oscillator: for 0< 𝛽 ≤1, there is an overdamped motion and no oscillation is observed; for 1< 𝛽 < 2, which is the most interesting regime and the one we will focus from here on, there is a motion reminiscent of a damped pendulum, namely, for short times one observes oscillatory motion with decaying amplitude that eventually changes to an overdamped behavior similar to that of a particle in a viscous medium. Finally, for 𝛽 > 2, an oscillatory motion with growing amplitude is observed, while, of course, for 𝛽= 2 the usual oscillatory behavior is recovered. For 1< 𝛽 ≤2, the Caputo derivative is defined as 𝜕𝛽𝜙(𝑥, 𝑡) 𝜕𝑡𝛽=1 𝛤(2 − 𝛽)∫𝑡 0 (𝑡−𝑠)1−𝛽[𝜕2𝜙(𝑥, 𝑡) 𝜕𝑡2]𝑡=𝑠 d𝑠(2.2) Notice that the choice of Caputo derivative instead of, e.g., the Riemann–Liouville derivative is motivated by the fact that we know the initial conditions for integer derivatives. In the next section, we study the evolution of the breather solution (1.2) for 𝑡= 0 as initial condition 𝜙(𝑥, 0) = 𝜓(𝑥)for (2.1). Moreover, in what follows, we set 𝜕𝑡𝜓(𝑥, 0) = 0. Numerical integration is performed by means of the multiterm MT_FPE_PI12_PC algorithm developed in Ref. 41. 3. Numerical results Let us start by considering the time-evolution of a breather for different values of 𝛽when 𝜔= 0.8is fixed. To this end, we take as initial condition 𝜓(𝑥, 0) = 𝜙br (𝑥, 0) in (1.2), i.e. 𝜙(𝑥, 0) = 4 arctan (√1 − 𝜔2 𝜔cosh(√1 − 𝜔2𝑥))(3.1) Fig. 1 shows the time evolution of such solutions for different values of 𝛽by means of a density plot; in this case, the displacements (rather than an energy density type quantity) are plotted. There are two different effects observed in this evolution. On the one hand, the breather deforms leading to the appearance of secondary ‘‘humps’’; on the other hand, the breather oscillations are damped. Indeed, as 𝛽 decreases, the breather survives only for a few oscillations (or, roughly, a single one as 𝛽becomes 1.5, see the bottom right panel of the figure). The left panel of Fig. 2 shows the deformation of the breather for 𝛽= 1.9by means of several snapshots. Notice that the maximal profile Partial Dierential Equations in Applied Mathematics 11 (2024) 100807 2 T. Bountis et al. Fig. 1. Space–time evolution of the breather for 𝜔= 0.8and 𝛽= 2 (top left), 𝛽= 1.99 (top right), 𝛽= 1.9(bottom left) and 𝛽= 1.5(bottom right). The rapid decay of the breather profile is clearly visible, especially in the latter two panels. Fig. 2. (Left panel) Breather deformation and appearance of secondary ‘‘humps’’ for 𝛽= 1.9and 𝑡= 0 (black), 𝑡= 200 (green), 𝑡= 700 (blue) and 𝑡= 950 (red). The amplitude of the solutions 𝜙(𝑥, 𝑡)decay with time but they are normalized to one in order to compare the evolution. (Right panel) Amplitude decay of the central node for 𝛽= 2 (gray), 𝛽= 1.99 (red), 𝛽= 1.9(blue) and 𝛽= 1.5(green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) amplitude has been rescaled to 1in this panel, so as to showcase the emergence of lateral (secondary) ‘‘humps’’. A more quantitative representation of the damping of the breather oscillations can be seen in the right panel of Fig. 2, which shows 𝜙(0, 𝑡) for several values of 𝛽. As is evident, fractionality modifies not only the amplitude of the oscillations, but also their frequency 𝜔which tends to unity, as the oscillation amplitude becomes so small that the linear regime is approached. This change in the frequency depends on 𝑥, implying loss of coherence of the breather. A natural byproduct of the latter is the complete disintegration of the structure observed in Fig. 1. Another interesting feature of the decay process of our fractional model is the existence of two different regimes. If we think of 𝜙(𝑥, 𝑡) as the deformation of a viscoelastic medium, so that it is elongated (compressed) when 𝜙 > 0(𝜙 < 0), we observe that for short times there are oscillations of the medium (i.e., an elastic regime) around a non-zero value of 𝜙, see left panels of Fig. 3 for two different values of 𝛽. After a time 𝑡𝑠, a regime shift can be observed as the medium remains compressed (𝜙 < 0for 𝑡 > 𝑡𝑠) and this compression relaxes to zero when 𝑡→∞, i.e., we are led to a viscous regime. This regime can be visualized in the right panels of Fig. 3, which are magnifications of the left panels for this regime. This decompression takes the form of oscillations for values of 𝛽close to 2(top right panel), and becomes more monotonic as one deviates further from this limit (bottom right panel). Next, Fig. 4 shows the dependence of 𝑡𝑠with respect to 𝛽for fixed 𝜔= 0.8. The value of 𝑡𝑠has been defined as the minimum value of 𝑡for which 𝜙(0, 𝑡)<0whenever 𝑡>𝑡𝑠, that is, the time at which the medium remains compressed and starts relaxing to zero. As can be expected, when the integer limit of 𝛽= 2 is approached, 𝑡𝑠tends to ∞, since in the nonfractional model this distinction of regimes does not exist. For Partial Dierential Equations in Applied Mathematics 11 (2024) 100807 3 T. Bountis et al. Fig. 3. Evolution of 𝜙(0, 𝑡)for 𝜔= 0.8, and 𝛽= 1.9(top panels) and 𝛽= 1.7(bottom panels). The system oscillates around zero for a certain time interval, until 𝑡=𝑡𝑠, when the system remains compressed and relaxes to zero. Right panels are zooms of left panels for 𝑡>𝑡𝑠so that the regime shifts can be observed. For 𝛽= 1.9, we find 𝑡𝑠= 120.98 and for 𝛽= 1.7,𝑡𝑠= 20.67.. Fig. 4. Shift from the elastic to the viscous regime dependence on 𝛽. 𝛽 < 2, this measure yields a quantitative diagnostic of the deviation from the benchmark scenario. We would like to stress that this behavior is different from that of a damped oscillator (or a breather in an elastic medium with losses), where the system oscillates around 0for every value of 𝑡. In fact, we have observed that this ‘‘viscoelastic’’ behavior is not only present for breathers, but also for isolated oscillators either linear or nonlinear. This strongly suggests that the effect of fractionality bears some qualitative differences in comparison to introducing (regular, linear) damping to the sine-Gordon equation with 𝛽= 2. Once we have qualitatively identified the dynamics of breathers, we proceed to analyze in more detail the observed decay in the elastic regime (i.e., for 𝑡<𝑡𝑠). In Fig. 5 we fit the extrema of the oscillation amplitude of 𝜙(0, 𝑡)to an exponential, i.e. 𝜙(0, 𝑡) ∼ 𝑎𝑒𝑏𝑡, with 𝜏= −1∕𝑏being the characteristic decay time. This figure also shows the dependence of 𝜏with respect to the fractionality 𝛽for two values of the breather frequency 𝜔. One can see that the dependence of 𝜏(𝛽)is qualitatively similar for both values of 𝜔. Naturally, also 𝜏→∞when 𝛽→2, as one approaches the classical (and Hamiltonian) sine-Gordon equation. If we now vary the frequency 𝜔and fix 𝛽= 1.9, we obtain a surprisingly non-monotonic behavior concerning the dependence of 𝜏on 𝜔(see bottom panel of Fig. 5). Fractionality seems to be more efficient as a damping ‘‘mechanism’’ for certain frequencies of the breather. Specifically, we obtain a minimum of 𝜏close to 𝜔= 0.7. A similar trend is found for other values of 𝛽. Despite the fact that the relevant variation in 𝜏is small, we do not have a ‘‘structural’’ explanation of this effect and attribute it, at least in part, to the particular functional form of the breather. In order to study if the previous observations related to damping are particular to fractional breathers, we now analyze the breather dynamics in connection to two fractional one-degree-of-freedom systems: a linear (harmonic oscillator) and a nonlinear one (pendulum). The latter one is of the form: d𝛽𝑢(𝑡) d𝑡𝛽+ sin(𝑢(𝑡)) = 0.(3.2) Our initial conditions for this system are taken as 𝑢(0) = 𝑢0,𝑢(0) = 0, where the oscillating amplitude 𝑢0is related to the oscillation frequency Partial Dierential Equations in Applied Mathematics 11 (2024) 100807 4 T. Bountis et al. Fig. 5. (Left panel) Evolution of 𝜙(𝑥, 0) for 𝜔= 0.8and 𝛽= 1.9. The envelope of the oscillations is close to an exponential: 𝑎𝑒𝑏𝑡, with 𝑎= 2.6598,𝑏= −0.1028 and 𝑟2= 0.997. (Right panel) Characteristic decay time 𝜏for 𝜔= 0.4and 𝜔= 0.8with respect to 𝛽. (Bottom panel) Decay times for different values of the frequency for fixed 𝛽= 1.9. The black line is just a fit to a cubic spline interpolation to guide the view. 𝜔through 𝜔=𝜋 2𝐾(𝑚)(sin2(𝑢0∕2)) ,(3.3) 𝐾(𝑚)being the complete elliptic integral of the first kind with parameter 𝑚. Thus, 𝑢0can be calculated for any desired value of the frequency. In the small amplitude limit, the fractional pendulum can be described by a fractional harmonic oscillator: d𝛽𝑢HO(𝑡) d𝑡𝛽+𝜔𝛽𝑢HO(𝑡)=0.(3.4) In this case, the solution is known and can be expressed as a series expansion of Mittag-Leffler functions 𝐸𝛽(𝑧),42 𝑢HO(𝑡) = 𝑢0𝐸𝛽[−(𝜔𝑡)𝛽], 𝐸𝛽(𝑧) = ∞ ∑ 𝑛=0 𝑧𝑛 𝛤(𝛽𝑛 + 1) .(3.5) Unlike the pendulum, in the harmonic oscillator, the frequency and the decay of oscillations are not affected by the initial amplitude. Thus, we can fix the initial condition to (𝑢0= 1,𝑢0= 0), without loss of generality. Interestingly, both the pendulum and the harmonic oscillator exhibit dynamics similar to the breather regarding their decay behavior. Fig. 6 compares the decay times of the breather, pendulum and harmonic oscillator with frequency 𝜔= 0.8with respect to 𝛽. It is important to observe that this decay time is shortest for the case of the breather, which suggests that the role of the additional degrees of freedom is to lead the solutions to a faster ‘‘decoherence’’. Furthermore, the nonlinear pendulum model is more rapidly decaying than the harmonic oscillator, suggesting that the nonlinearity enhances the dissipative dynamics in this context. On the right panel of Fig. 6, the decay time with respect to the frequency is shown for 𝛽= 1.9in the case of the pendulum. One can see that, contrary to the breather case, this dependence is monotonic. This is also the case for the harmonic oscillator. Once again, as indicated above, the variations of the relevant time scale 𝜏are rather small, yet there is a clear distinction of the ‘‘collective’’ sine-Gordon model in comparison to the single degree of freedom systems for sufficiently large frequencies, i.e., for 𝜔 > 0.7 (where the minimum of the quantity 𝜏occurs for the sine-Gordon case). 4. The effect of anti-damping terms Finally, we discuss the possibility of adding an anti-damping term to (2.1) to avoid the effects of fractionality, i.e., the emergence of exponential damping, and secondly, the appearance of a regime shift that separates the evolution of the system in a viscoelastic and a viscous phase. We do not expect to avoid the latter effect, but counter-acting the exponential damping with such an anti-damping term is of interest in its own right. To this end we study the following system: 𝜕𝛽𝜙(𝑥, 𝑡) 𝜕𝑡𝛽−𝜕2𝜙(𝑥, 𝑡) 𝜕𝑥2+ sin(𝜙(𝑥, 𝑡)) + 𝛾𝜕𝜙(𝑥, 𝑡) 𝜕𝑡 = 0,(4.1) where the anti-damping corresponds to the last term, with 𝛾 < 0. We observe that from a certain value on, the damping induced by the fractionality disappears and oscillations are maintained or amplified. For 𝛽= 1.9and 𝜔= 0.8, this value is 𝛾𝑐≈ −0.1245. In the top left panel of Fig. 7 we can see trajectories in phase space (𝜙(0, 𝑡), 𝜕𝑡𝜙(0, 𝑡)) for values of 𝛾larger and smaller than 𝛾𝑐. It is clear that the case below 𝛾𝑐still exhibits damping, while the other one slightly above counter-balances this effect and leads to some residual, weak pumping. However, as shown in the top right panel of Fig. 7, the antidamping is not able to restore the initial frequency which was modified by Partial Dierential Equations in Applied Mathematics 11 (2024) 100807 5 T. Bountis et al. Fig. 6. (Left panel) Characteristic decay time 𝜏with respect to 𝛽for 𝜔= 0.8in the breather, pendulum and harmonic oscillator cases. Notice that for the HO, 𝜔=𝜔𝛽 0= 0.8(Right panel) Decay times for different values of the frequency for fixed 𝛽= 1.9in the pendulum case. The black line is just a fit to a cubic spline interpolation to guide the view. the fractionality. In fact, we observe that for the blue line (𝛽= 2, 𝛾= 0), the frequency is 𝜔= 0.8. When the fractionality is introduced, which corresponds to the black line (𝛽= 1.9,𝛾= 0), the frequency is modified to 𝜔= 1. Finally, when we add the antidamping term, which is represented by the red line (𝛽= 1.9,𝛾= −0.125), oscillations no longer decay, in fact, they are amplified, but the frequency is changed to approximately 𝜔= 0.75. The frequency spectrum is depicted in the bottom left panel of Fig. 7. As it can be seen, there is a small peak at 3𝜔=2.25, which corresponds to one of the harmonics of the main frequency peak. For other values of 𝑥≠0, close to the central node, we obtain the same frequency spectrum. These facts suggest that a frequency-shifted breather is generated. This needs to be explored in the future by means of fixed point algorithms. It is also interesting to notice that for 𝛾 > 𝛾𝑐the dynamics of the system becomes more complex and as the amplitude increases the system may jump to a different attractor. Thus, the oscillations seen in Fig. 7 for 𝛾= −0.125 are only a transient phenomenon. At longer times, as is it observed in the bottom right panel, the structure breaks down into a pair of kinks that subsequently move outward from 𝑥= 0. We conclude that fractionality perturbs the system in a way that the antidamping term cannot fully amend. 5. Conclusions and future challenges In a recent paper,37 a subset of the present authors considered a generalized nonlinear wave equation in multiple dimensions which extends well-known conservative models of mathematical physics, such as the sine-Gordon equation, from their classical to their fractional form. In that work, fractional temporal derivatives of the Caputo type and spatial Riesz-fractional derivatives were utilized. In the same work,37 the existence and uniqueness of solutions was proved for this model using a fixed-point theorem, and it was verified that a consistent discretization of the conserved quantity is also preserved through time. Our main purpose in the present paper is to emphasize the dissipative nature of the Caputo time-fractional derivative, a feature that was present but not further detailed in the earlier work of Ref. 37. This effect is present in numerous of the results discussed here for the case of a time-fractional sine-Gordon equation and order of differentiation 𝛽 < 2. It is important to note that the sine-Gordon equation remains a conservative model, even when space-fractional derivatives of the Riesz type are considered. Evidently, therefore, it is the presence of the Caputo time-fractional derivatives that turns this system into a dissipative model. Analogous results have also been observed in the literature43–48 mostly in cases where ordinary (rather than partial) differential equations are studied. Relevant explicit results are available, e.g., in the case of linear harmonic oscillator (e.g., in the work of Ref. 45) showcasing the decreasing amplitude of the relevant model. Here, we have taken these results further, not only developing spatial and temporal diagnostics for the evolution of the breather, but also offering comparisons with the single DOF models, illustrating the role of the nonlinearity and of the coupling to the additional degrees of freedom (in comparison to a single pendulum) towards decreasing the lifetime of the pertinent breathing oscillations. Furthermore, we have considered a ‘‘counter-action’’ mechanism through an anti-damping, and have clarified which aspects of the evolution we can hope to ‘‘restore’’ (such as, e.g., the amplitude) and which remain modified (such as, e.g., the frequency). Thus, it is clear that a deeper analysis of this ‘‘dissipative’’ phenomenon needs to be carried out in future works concerning the solutions of conservative systems of fractional differential equations, whose derivatives with respect to time are of the Caputo type. Furthermore, both the present work and that of Ref. 37 offer additional insights for possible future studies. In particular, in the presence of anti-damping herein, we have observed the breakdown of the breather structure towards a kink–antikink pair. On the other hand in Ref. 37, it was observed that the combined action of the Caputo time derivative and the Riesz spatial derivative (when suitably deviating from the sine- Gordon limit) can lead to nearly stationary kink–antikink-like bound states. Both of these results, but also the topological nature of the kink structures, suggest the particular interest and relevance of the study of such topological states and their multi-kink generalizations. Indeed, it is important to recall that, unlike the breather, the kink cannot be subject to destruction (although it can be, e.g., subject to deceleration and stopping, akin to what happens in discrete variants of the sine- Gordon model Ref. 49). This is a natural next step for our considerations which is presently under study and which will be reported in future publications. Funding J.E.M.-D.: The present work reports on a set of final results of the research project ‘‘Conservative methods for fractional hyperbolic systems: analysis and applications’’, funded by the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S- 45928. This material is based upon work supported by the U.S. National Science Foundation under the awards PHY-2110030 and DMS-2204702 (PGK). J.C.-M. acknowledges support from the EU (FEDER program 2014–2020) through MCIN/AEI/10.13039/501100011033 (under the projects PID2020-112620GB-I00 and PID2022-143120OB-I00). J.C. acknowledges that this work has been supported by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF, EU) under Project No. PID2019105554GB-I00 (MCIN/AEI/ 10.13039/ 501100011033). Partial Dierential Equations in Applied Mathematics 11 (2024) 100807 6 T. Bountis et al. Fig. 7. (Top left panel) Trajectories in phase space for 𝑥= 0 and values of 𝛾larger and smaller than 𝛾𝑐. The blue dot denotes the common initial condition. At the beginning both trajectories follow the same path, but after a while, one is amplified while the other one is damped. (Top right panel) Time evolution for the breather with frequency 𝜔= 0.8 without fractionality (blue line), with fractionality (black line) and with fractionality and antidamping (red line). Fractionality induces both damping and a change in the frequency. The antidamping may restore the amplitude but not the frequency. (Bottom left panel) Frequency spectrum for 𝜙(0, 𝑡)with 𝛽= 1.9and 𝛾= −0.125. (Bottom right panel) Space–time evolution of the breather for 𝛽= 1.9with 𝛾= −0.125. For a transient time, oscillations with slightly increasing amplitude are recovered, before the configuration eventually breaks up into a pair of moving kinks. In all panels, 𝛽= 1.9,𝜔= 0.8. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) CRediT authorship contribution statement Tassos Bountis: Conceptualization, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing. Julia Cantisán: Formal analysis, Methodology, Software, Validation, Writing – original draft, Writing – review & editing. Jesús Cuevas- Maraver: Formal analysis, Methodology, Software, Validation, Writing – original draft, Writing – review & editing. J.E. Macías-Díaz: Conceptualization, Validation, Writing – review & editing. Panayotis G. Kevrekidis: Formal analysis, Methodology, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing. 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