Exploring the interplay between fractionality and PT symmetry in magnetic metamaterials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917948511223808 |
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| author | Molina, Mario I. |
| author_facet | Molina, Mario I. |
| contents | We study a nonlinear magnetic metamaterial modeled as a split-ring resonator array, where the standard discrete laplacian is replaced by its fractional form. We find a closed-form expression for the dispersion relation as a function of the fractional exponent s and the gain/loss parameter γ and examine the conditions under which stable magneto-inductive waves exist. The density of states is computed in closed form and suggests that the main effect of fractionality is the flattening of the bands, while gain/loss increase tends to reduce the bandgaps. The spatial extent of the modes for a finite array is computed by means of the participation ratio R, which is also obtained in closed form. For a fixed fractionality exponent, an increase in gain/loss γ decreases the overall R, from the number of sites N towards N/2 at large γ. The nonlinear dynamics of the average magnetic energy on an initial ring during a cycle shows a monotonic increase with γ, and it is qualitatively similar for all fractional exponents. This is explained as mainly due to the interplay of nonlinearity and PT symmetry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exploring the interplay between fractionality and PT symmetry in magnetic metamaterials Molina, Mario I. Pattern Formation and Solitons We study a nonlinear magnetic metamaterial modeled as a split-ring resonator array, where the standard discrete laplacian is replaced by its fractional form. We find a closed-form expression for the dispersion relation as a function of the fractional exponent s and the gain/loss parameter γ and examine the conditions under which stable magneto-inductive waves exist. The density of states is computed in closed form and suggests that the main effect of fractionality is the flattening of the bands, while gain/loss increase tends to reduce the bandgaps. The spatial extent of the modes for a finite array is computed by means of the participation ratio R, which is also obtained in closed form. For a fixed fractionality exponent, an increase in gain/loss γ decreases the overall R, from the number of sites N towards N/2 at large γ. The nonlinear dynamics of the average magnetic energy on an initial ring during a cycle shows a monotonic increase with γ, and it is qualitatively similar for all fractional exponents. This is explained as mainly due to the interplay of nonlinearity and PT symmetry. |
| title | Exploring the interplay between fractionality and PT symmetry in magnetic metamaterials |
| topic | Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2503.06289 |