Lax pair, Darboux transformation and rogue waves for the three-coupled fourth-order nonlinear Schrodinger system in an alpha helical protein

Complexes of proteins are central to certain cellular processes. Investigated in this paper is the three-coupled fourth-order nonlinear Schrodinger system, which is used for describing the alpha helical proteins with interspine coupling at the fourth order. With respect to the three-component amplitudes of the molecular excitation, we derive a Lax pair and construct the corresponding Nth-order Darboux transformation, where N is a positive integer. Three types of the Nth-order rogue wave solutions are obtained with the help of the matrix analysis method. The first-order rogue waves with each component containing one, two or three rogue waves are derived. We observe that the width of the first-order vector rational rogue wave along the distance axis increases with the value of the lattice parameter increasing, while the width of the first-order vector rational rogue wave along the time axis decreases with the value of the lattice parameter increasing. We present the second-order vector rational rogue waves with each component constituted by five, seven or nine rogue waves. Vector semi-rational rogue waves display the coexistence of the rogue waves and line/Y-shaped breathers.