Localization and global dynamics in the long-range discrete nonlinear Schrödinger equation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909780500545536 |
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| author | Choi, Brian Marstaller, Austin Aceves, Alejandro |
| author_facet | Choi, Brian Marstaller, Austin Aceves, Alejandro |
| contents | We study localization, pinning, and mobility in the fractional discrete nonlinear Schrödinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite and offsite stationary profiles; their asymptotic validity, orbital stability of onsite solutions, and $\ell^2$ proximity to the exact lattice solutions are established. Using the explicit construction of localized states, it is shown that the spatial tail behavior is algebraic for all $α$ > 0. The Peierls-Nabarro barrier (PNB) is computed, and the parameter regimes are identified where it nearly vanishes; complementary numerical simulations explore mobility/pinning across parameters and exhibit scenarios consistent with near-vanishing PNB. We also analyze modulational instability of plane waves, locate instability thresholds, and discuss the role of nonlocality in initiating localization. Finally, we establish small-data scattering, and quantify how fDNLS dynamics approximates the nearest-neighbor DNLS on bounded times while exhibiting distinct global behavior for any large $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11395 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Localization and global dynamics in the long-range discrete nonlinear Schrödinger equation Choi, Brian Marstaller, Austin Aceves, Alejandro Classical Analysis and ODEs Analysis of PDEs 34A08, 34A12, 34A34, 37K45, 37K60, 78M35 We study localization, pinning, and mobility in the fractional discrete nonlinear Schrödinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite and offsite stationary profiles; their asymptotic validity, orbital stability of onsite solutions, and $\ell^2$ proximity to the exact lattice solutions are established. Using the explicit construction of localized states, it is shown that the spatial tail behavior is algebraic for all $α$ > 0. The Peierls-Nabarro barrier (PNB) is computed, and the parameter regimes are identified where it nearly vanishes; complementary numerical simulations explore mobility/pinning across parameters and exhibit scenarios consistent with near-vanishing PNB. We also analyze modulational instability of plane waves, locate instability thresholds, and discuss the role of nonlocality in initiating localization. Finally, we establish small-data scattering, and quantify how fDNLS dynamics approximates the nearest-neighbor DNLS on bounded times while exhibiting distinct global behavior for any large $α$. |
| title | Localization and global dynamics in the long-range discrete nonlinear Schrödinger equation |
| topic | Classical Analysis and ODEs Analysis of PDEs 34A08, 34A12, 34A34, 37K45, 37K60, 78M35 |
| url | https://arxiv.org/abs/2309.11395 |