The Effect of On-Site Potentials on Supratransmission in One-Dimensional Hamiltonian Lattices

We investigated a class of one-dimensional (1D) Hamiltonian <i>N</i>-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one end of the lattice and an absorbing boundary on the other, we studied the phenomenon of supratransmission and its dependence on two ranges of interactions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, as the effect of the on-site potential terms of the Hamiltonian varied. In previous works, we studied the critical amplitude <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi mathvariant="sans-serif">Ω</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> at which supratransmission occurs, for one range parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, and showed that there was a sharp threshold above which energy was transmitted in the form of large-amplitude nonlinear modes, as long as the driving frequency <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula> lay in the forbidden band-gap of the system. In the absence of on-site potentials, it is known that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi mathvariant="sans-serif">Ω</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> increases monotonically the longer the range of interactions is (i.e., as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>⟶</mo><mn>0</mn></mrow></semantics></math></inline-formula>). However, when on-site potential terms are taken into account, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>s</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi mathvariant="sans-serif">Ω</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> reaches a maximum at a low value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> that depends on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula>, below which supratransmission thresholds <i>decrease</i> sharply to lower values. In this work, we studied this phenomenon further, as the contribution of the on-site potential terms varied, and we explored in detail their effect on the supratransmission thresholds.