Davydov soliton evolution in temperature gradients driven by hyperbolic waves

In the present work the evolution of a Davydov soliton in an inhomogeneous medium will be considered. The Zakharov system of equations, which describes this soliton, consists of a perturbed non-linear Schrödinger (NLS) type equation plus a forced wave equation. This system is not exactly integrable for a homogeneous medium and its Lagrangian is non-local. It has recently been shown that this type of soliton has a long enough lifetime, even for non-zero temperature, so as to be a possible mechanism for the transfer of energy along an α helix. In the present work, the effect of temperature inhomogeneities on the behaviour of this soliton will be studied. As the soliton propagates through such an inhomogeneity, both dispersive and non-dispersive waves are generated. The stability of the soliton to this radiation is studied. The evolution of the Davydov soliton solution of the Zakharov equations in an inhomogeneous medium will be studied using an approximate method based on averaged conservation laws, which results in ordinary differential equations for the pulse parameters. It is shown that the inclusion of the effect of the dispersive radiation shed by the soliton for the NLS equation and the non-dispersive (hyperbolic) radiation shed by the soliton for the forced wave equation is vital for an accurate description of the evolution of the Davydov soliton. It is found that the soliton is stable even in the presence of hyperbolic radiation and that the temperature gradients have significant effects on the propagation of the soliton, even to the extent of reversing its motion.