Modulational instability is a direct way by which localized structures emerge in nonlinear systems. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in α-helix proteins with three spines. Through numerical simulations, trains of pulses are found and confirm our analytical predictions. The presence of higher-order interactions between adjacent spines tends to suppress the formed localized structures for erratic ones to emerge. These erratic structures are highly localized and rather reinforce the idea that the energy to be used in metabolic processes is rather confined to specific regions for its efficiency.
A three-stranded α-helix protein chain model with long-rangeinteractions, whose dynamic, in the continuum limit approximation isgoverned by three coupled nonlinear Schr\"{o}dinger equation typesis investigated. Applying the trial equation method, two- andfour-bright solitary wave solutions are constructed to explain theexciton dynamic through the protein chains. Intensive numericalsimulations allow to check the influence of key system parameters onthe dynamic of the solutions. It appears that nonlinear couplings,long-range interactions, asymmetric molecular vibrations, and thewave's velocity deeply alter the widths and amplitudes of oursolitary waves, two salient features which mean that the abovementioned parameters have a major impact on bioenergy transport inprotein chains. In the same vein, the number of solitary waves whichpropagate through the protein chain may be controlled by either thewave velocity, the asymmetric molecular vibrations strength or thelong-range interaction strength provided that values of the latterparameters are set below or above critical values identified.
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