General Higher-Order Breather and Hybrid Solutions of the Fokas System

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schrodinger (NLS) equation (Eq. (2), Inverse Problems 10 (1994) L19-L22). By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of n-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits a range of interesting dynamics. For high-order breather, it has more rich dynamical behaviors. The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively. By taking a long waves limit of a part of exponential functions in f and g appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.