Modulational instability and homoclinic orbit solutions in vector nonlinear Schrödinger equation

• Published in: Communications in nonlinear science & numerical simulation 2019-06, Vol.72, p.449-471
• Main Authors: Ling, Liming, Zhao, Li-Chen
• Format: Article
• Language: English
• Subjects: Akhmediev breathers; Algebra; Breathers; General multi-high-order rogue wave; Modulational instability analysis; Nonlinear equations; Nonlinear systems; Orbital stability; Schrodinger equation; Stability analysis; Vector nonlinear Schrödinger equation
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.01.008

•The Akhmediev breather (AB) solutions for the vector nonlinear Schrödinger equation (VNLSE) are constructed by the Darboux transformation.•We develop a method to determine the spectral curve where the AB solutions are located.•The high order rogue waves (RWs), multi-RWs and multi-high order RWs are derived by Darboux transformation.•We find the dispersion equation for the MI is equivalent with determined equation for ABs or RWs. Modulational instability has been used to explain the formation of breathers and rogue waves qualitatively. In this paper, we show modulational instability can be used to explain the structure of them in a quantitative way. In the first place, we develop a method to derive general forms for Akhmediev breathers, rogue waves and their multiple or high order ones in a N-component nonlinear Schrödinger equations. The existence condition for each pattern is clarified clearly with a compact algebraic equation. Moreover, we show that the existence condition of ABs and RWs is consistent with the dispersion relation of the linear stability analysis on the background solution. The results further deepen our understanding on the quantitative relations between modulational instability and homoclinic orbits solutions.