The Dynamics of Pole Trajectories in the Complex Plane and Peregrine Solitons for Higher-Order Nonlinear Schrödinger Equations: Coherent Coupling and Quintic Nonlinearity

Abstract

The Peregrine soliton is an exact, rational, and localized solution of the nonlinear Schrödinger equation and is commonly employed as a model for rogue waves in physical sciences. If the transverse variable is allowed to be complex by analytic continuation while the propagation variable remains real, the poles of the Peregrine soliton travel down and up the imaginary axis in the complex plane. At the turning point of the pole trajectory, the real part of the complex variable coincides with the location of maximum height of the rogue wave in physical space. This feature is conjectured to hold for at least a few other members of the hierarchy of Schrödinger equations. In particular, evolution systems with coherent coupling or quintic (fifth-order) nonlinearity will be studied. Analytical and numerical results confirm the validity of this conjecture for the first- and second-order rogue waves.