Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials
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Date
2019-04-03
Authors
Mac´ıas-D´ıaz, Jorge E.
Bountis, Anastasios
Christodoulidi, Helen
Journal Title
Journal ISSN
Volume Title
Publisher
Mathematics in Engineering
Abstract
We consider a family of 1-dimensional Hamiltonian systems consisting of a large number
of particles with on-site potentials and global (long range) interactions. The particles are initially at rest at the equilibrium position, and are perturbed sinusoidally at one end using Dirichlet data, while at the other end we place an absorbing boundary to simulate a semi-infinite medium. Using such a lattice with quadratic particle interactions and Klein-Gordon type on-site potential, we use a parameter 0 ≤ α < ∞as a measure of the “length” of interactions, and show that there is a sharp threshold above which energy is transmitted in the form of large amplitude nonlinear modes, as long as driving frequencies
Ω lie in the forbidden band-gap of the system. This process is called nonlinear supratransmission and is investigated here numerically to show that it occurs at higher amplitudes the longer the range of interactions, reaching a maximum at a value α = αmax . 1.5 that depends on Ω. Below this αmax supratransmission thresholds decrease sharply to values lower than the nearest neighbor α = ∞ limit. We give a plausible argument for this phenomenon and conjecture that similar results are present in related systems such as the sine-Gordon, the nonlinear Klein-Gordon and the double sine-Gordon type.
Description
Keywords
nonlinear supratransmission, globally interacting systems, on-site potentials
Citation
Jorge E. Mac´ıas-D´ıaz, Anastasios Bountis,and Helen Christodoulidi (2019) Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials. Mathematics in Engineering.Volume: 1, Issue: 2, Pages: 343-358