We investigate a family of one--dimensional (1D) Hamiltonian $N-$particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on--site terms of the sine--Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. Using reliable finite--difference discretization schemes, we establish the existence of nonlinear supratransmission for both short--range and long--range interactions, and demonstrate that the presence of quadratic interactions is {\it not} necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on--site potentials is also not crucial for the system to present supratransmission.

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