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Abstract In this paper, we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations. After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation. We then compute, both analytically and numerically, its travelling wave and periodic travelling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one. 

In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order β of the temporal derivative to that of a Caputo fractional type and found that, for 1<β<2, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order α on the spatial derivative. Here, depending on whether this order was below or above the harmonic value α=2, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.