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• W3100283516 abstract "We study a class of discrete focusing nonlinear Schrödinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls–Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the case of nearest neighbor interactions to a large class of nonlocal short-range and long-range interactions. The appearance of distinct onsite and offsite states is a consequence of the breaking of continuous spatial translation invariance. The PNB plays a role in the dynamics of energy transport in such nonlinear Hamiltonian lattice systems. Our class of nonlocal interactions is defined in terms of coupling coefficients, J m , where $${minmathbb{Z}}$$ is the lattice site index, with $${J_msimeq m^{-1-2s}, sin[1,infty)}$$ and $${J_msim e^{-gamma|m|}, s=infty, gamma > 0,}$$ (Kac–Baker). For $${sge1}$$ , the bifurcation is seeded by solutions of the (effective/homogenized) cubic focusing nonlinear Schrödinger equation (NLS). However, for $${1/4 < s < 1}$$ , the bifurcation is controlled by the fractional nonlinear Schrödinger equation, FNLS, with $${(-Delta)^s}$$ replacing $${-Delta}$$ . The proof is based on a Lyapunov–Schmidt reduction strategy applied to a momentum space formulation. The PN barrier bounds require appropriate uniform decay estimates for the discrete Fourier transform of DNLS discrete solitary waves. A key role is also played by non-degeneracy of the ground state of FNLS, recently proved by Frank, Lenzmann and Silvestrie." @default.
• W3100283516 created "2020-11-23" @default.
• W3100283516 creator A5015779479 @default.
• W3100283516 creator A5074916646 @default.
• W3100283516 date "2017-02-14" @default.
• W3100283516 modified "2023-09-29" @default.
• W3100283516 title "Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls–Nabarro Barrier" @default.
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• W3100283516 doi "https://doi.org/10.1007/s00220-017-2839-4" @default.
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