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

Ning N Peng,Kwok W Chow,Tin L Chiu

doi:10.3389/fphy.2020.581662

Abstract

The Peregrine breather is an exact, rational and localized solution of the nonlinear Schrodinger 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 breather 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 displacement of the rogue wave in physical space. This feature is conjectured to hold for at least a few other members of the hierarchy of Schrodinger 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.

Highlights

The Peregrine soliton is an exact, rational solution of the nonlinear Schrödinger equation (NLSE) [1]
Our goal is to provide still another perspective, namely, utilizing the dynamics of pole trajectories in the complex plane to elucidate the properties of rogue waves [8,9,10]
The maximum height of the rogue wave (Peregrine soliton) in the physical space occurs at the location x = 0, which is the real part of the turning point in the pole trajectories in the complex x plane

Summary

INTRODUCTION

The Peregrine soliton is an exact, rational solution of the nonlinear Schrödinger equation (NLSE) [1]. The NLSE is widely used to model wave packet dynamics in various disciplines in physical science, e.g., fluid mechanics and optics [2, 3]. A correlation on the locations of maximum height of a rogue wave in the physical space and the real parts of the poles in the complex plane is proposed. How this conjecture can be verified for the more complicated case of coherently coupled Schrödinger equations is elucidated (Section 3). We discuss physical insights and draw conclusions (Section 5)

THE PEREGRINE SOLITON

Verification of the Conjecture

Rogue Waves With “Double Peaks”

A SCHRÖDINGER EQUATION WITH QUINTIC NON-LINEARITY

DISCUSSIONS AND CONCLUSIONS

DATA AVAILABILITY STATEMENT