Modulation instability gain and Modulated wave shape incited by the acoustic longitudinal vibrations in Molecular Chain Model

The propagation of the modulated wave pattern and its instability in nonlinear medium is an important study from where experimental results are obtained. In this work we use the Nonlinear Schr\"{o}dinger Equation (NLSE) with saturation nonlinearity which describes the solitonic waves in molecular chain model. We set our study on the effectiveness of the acoustic longitudinal velocity which is related to the acoustic longitudinal vibration to investigate the behavior of the modulated wave patterns as well as the modulation instability (MI) gain. Its results that the acoustic longitudinal velocity creates instability zones, increases MI bands and shortens the transition regime of the modulated wave pattern. For long time of simulation and great enough value of the acoustic longitudinal velocity the modulated waves feature chaos-like motion. Alongside of these results the acoustic longitudinal velocity obvious itself as tool control of modulated wave as well as an energy source for the propagation of diverse modulated wave shaped in molecular chain.