Mathematics in Engineering, 2019, 1(2): 343-358. doi: 10.3934/mine.2019.2.343 Research article Special Issues

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Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials

Jorge E. Macías-Díaz, Anastasios Bountis, Helen Christodoulidi

1 Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Ags. 20131, Mexico
2 Department of Mathematics, Nazarbayev University, Kabanbay-Batyr 53, 010000 Astana, Republic of Kazakhstan
3 Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4 Athens, GR-11527, Greece
$^\dagger$This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics
Guest Editors: Simone Paleari; Tiziano Penati
Link: http://www.aimspress.com/newsinfo/1165.html

Received: , Accepted: , Published:

Special Issues: Hamiltonian Lattice Dynamics

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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 $0\leq\alpha<\infty$ 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 $\Omega$ 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 $\alpha=\alpha_{max} \lesssim 1.5$ that depends on $\Omega$. Below this $\alpha_{max}$ supratransmission thresholds decrease sharply to values lower than the nearest neighbor $\alpha=\infty$ 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.

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