The Dynamics and Evolution of Poles and Rogue Waves for Nonlinear Schrödinger Equations ^*

Abstract

Rogue waves are unexpectedly large deviations from equilibrium or otherwise calm positions in physical systems, e.g. hydrodynamic waves and optical beam intensities. The profiles and points of maximum displacements of these rogue waves are correlated with the movement of poles of the exact solutions extended to the complex plane through analytic continuation. Such links are shown to be surprisingly precise for the first order rogue wave of the nonlinear Schrödinger (NLS) and the derivative NLS equations. A computational study on the second order rogue waves of the NLS equation also displays remarkable agreements.

A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane

“…15 Preliminary numerical testing has been performed for the second order rogue waves of the nonlinear Schršdinger equation and the results are very encouraging. 15 The goal here is to conduct further numerical tests using a real evolution system and also asymmetric as well as symmetric rogue waves for complex-valued evolution equations. Such numerical tests can be justified from the following perspective.…”

“…The lowest order rogue wave is discussed in Section II, while the second order one is addressed in our earlier paper. 15 Here we focus on more complicated cases, namely, asymmetric rogue waves and the occurrence of such large amplitude modes for coupled systems.…”

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

“…We proposed a conjecture (Section 2) which will hopefully provide an important step in this direction [8][9][10]. If the transverse variable of the Schrödinger equation is allowed to be complex, the real parts of the pole trajectories at the turning points will be identical to or will closely correlate with the locations of maximum heights of the rogue wave in physical space.…”

“…Experimentally, the occurrence of rogue waves is realized through wave channels in hydrodynamics and fiber laser setting in optics [7]. 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].…”

Rotating spin-1/2 Bose–Einstein condensates in a gradient magnetic field with spin–orbit coupling

“…In spinorbit coupled spin-1/2 BECs, the competition between the intraspecies repulsion and the interspecies repulsion leads to a variety of ground-state phases, such as the plane wave phase, stripe phase and meron with dipole-dipole repulsion [16][17][18][19][20]. The dynamic properties of spin-orbit coupled spin-1/2 BECs are also studied, such as solitons, quantized vortices dynamics and Josephson dynamics [21][22][23][24][25][26][27][28][29].…”