No abstract

. (2017) A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane.Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (9). 091103. ISSN 1054-1500. DOIhttps://doi.org/10.1063/1.5001007 Link to record in KAR

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.

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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.