Davydov soliton in alpha helical proteins: higher order and discreteness effects

We consider the dynamics of Davydovʹs model of alpha helical proteins by including the effects of higher order molecular excitations and interactions and also the discreteness effect. In the continuum limit, the dynamics is found to be governed by a perturbed higher order non-linear Schrödinger equation which under appropriate limits reduces to the integrable cubic non-linear Schrödinger equation and also to an integrable higher order non-linear Schrödinger equation. A perturbation analysis of the equation shows that the higher order and discreteness effects do not change the velocity and amplitude of the Davydov soliton. An explicit form of the perturbed soliton solution has also been obtained.