Study on the generation mechanism of bright and dark solitary waves and rogue wave for a fourth-order dispersive nonlinear Schrödinger equation

In this paper, we study the generation mechanism of bright and dark solitary waves and rogue wave for the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation, which can not only model the nonlinear propagation and interaction of ultrashort pulses in the high-speed optical fiber transmission system, but also govern the nonlinear spin excitations in the onedimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole-dipole interaction. Firstly, via the phase plane analysis, we obtain both the homoclinic and heteroclinic orbits for the two-dimensional plane autonomous system reduced from the FODNLS equation. Further, we derive the bright and dark solitary wave solutions under the corresponding conditions, which reveals the relationship between the homoclinic (heteroclinic) orbit and solitary wave. Secondly, based on the exact first-order breather solution of the FODNLS equation over a nonvanishing background, we give the explicit expressions of group and phase velocities, and reveal that there exists a jump in both the velocities. Finally, in order to verify that the breather becomes a rogue wave at the jumping point, we obtain the first-order rogue wave solution by taking the limit of the breather solution at such point, which confirms the relationship of the generation of rogue wave with the velocity discontinuity.