dorsal/arxiv
View SchemaExcitation Thresholds for Nonlinear Localized Modes on Lattices
| Authors | Michael I. Weinstein |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9903001 |
| URL | https://arxiv.org/abs/patt-sol/9903001 |
| DOI | 10.1088/0951-7715/12/3/314 |
Abstract
Breathers are spatially localized and time periodic solutions of extended Hamiltonian dynamical systems. In this paper we study excitation thresholds for (nonlinearly dynamically stable) ground state breather or standing wave solutions for networks of coupled nonlinear oscillators and wave equations of nonlinear Schr\"odinger (NLS) type. Excitation thresholds are rigorously characterized by variational methods. The excitation threshold is related to the optimal (best) constant in a class of discr ete interpolation inequalities related to the Hamiltonian energy. We establish a precise connection among $d$, the dimensionality of the lattice, $2\sigma+1$, the degree of the nonlinearity and the existence of an excitation threshold for discrete nonlinear Schr\"odinger systems (DNLS). We prove that if $\sigma\ge 2/d$, then ground state standing waves exist if and only if the total power is larger than some strictly positive threshold, $\nu_{thresh}(\sigma, d)$. This proves a conjecture of Flach, Kaldko& MacKay in the context of DNLS. We also discuss upper and lower bounds for excitation thresholds for ground states of coupled systems of NLS equations, which arise in the modeling of pulse propagation in coupled arrays of optical fibers.
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"abstract": "Breathers are spatially localized and time periodic solutions of extended\nHamiltonian dynamical systems. In this paper we study excitation thresholds for\n(nonlinearly dynamically stable) ground state breather or standing wave\nsolutions for networks of coupled nonlinear oscillators and wave equations of\nnonlinear Schr\\\"odinger (NLS) type. Excitation thresholds are rigorously\ncharacterized by variational methods. The excitation threshold is related to\nthe optimal (best) constant in a class of discr ete interpolation inequalities\nrelated to the Hamiltonian energy. We establish a precise connection among $d$,\nthe dimensionality of the lattice, $2\\sigma+1$, the degree of the nonlinearity\nand the existence of an excitation threshold for discrete nonlinear\nSchr\\\"odinger systems (DNLS).\n We prove that if $\\sigma\\ge 2/d$, then ground state standing waves exist if\nand only if the total power is larger than some strictly positive threshold,\n$\\nu_{thresh}(\\sigma, d)$. This proves a conjecture of Flach, Kaldko\u0026 MacKay in\nthe context of DNLS. We also discuss upper and lower bounds for excitation\nthresholds for ground states of coupled systems of NLS equations, which arise\nin the modeling of pulse propagation in coupled arrays of optical fibers.",
"arxiv_id": "patt-sol/9903001",
"authors": [
"Michael I. Weinstein"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1088/0951-7715/12/3/314",
"title": "Excitation Thresholds for Nonlinear Localized Modes on Lattices",
"url": "https://arxiv.org/abs/patt-sol/9903001"
},
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