<p>In this paper, modulation instability and modulated wave patterns are developed in an anharmonic monoatomic chain. The linear stability with small perturbations is used to establish an expression for the modulation instability growth rate, which permits pointing out stable and unstable zones under the effects of the nearest-neighbor interaction. It is demonstrated that the variation of the anharmonic term can affect the modulation instability bands and reduce the unstable zone. Throughout the numerical simulation, we have pointed out rogue waves and different forms of the trains of pulse to confirm the existence of modulation instability. Using the semidiscrete multi-scale method, we have derived the nonlinear Schrödinger equation from where we have displayed the periodic structures and solitonic waves in the allowed phonon band. Using the nonlinear supratransmission phenomenon, we point out the localized waves and trains of pulse in the forbidden gap where the driven amplitude is considered above the supratransmission threshold. Chaotic behavior is observed to emerge within a specific range of propagation time. These findings may open new avenues for further research on delocalized waves in monoatomic chains.</p>