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

ABSTRACT

Complexes of proteins are central to certain cellular processes. Investigated in this paper is the three-coupled fourth-order nonlinear Schrödinger 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.

Keywords:

• Alpha helical protein
• three-coupled fourth-order nonlinear Schrödinger system
• rational and semi-rational rogue waves
• Darboux transformation

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been supported by the BUPT Excellent Ph.D. Students Foundation under No. CX2019201, by the National Natural Science Foundation of China under Grant Nos. 11772017 and 11805020, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.