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

Liu, T. Y., Chiu, T. L., Clarkson, Peter, Chow, K. W. (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. E-ISSN 1089-7682. (doi:10.1063/1.5001007)

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Abstract

Rogue waves of evolution systems are displacements which are localized in both space and time. The locations of the points of maximum displacements of the wave profiles may correlate with the trajectories of the poles of the exact solutions from the perspective of complex variables through analytic continuation. More precisely, the location of the maximum height of the rogue wave in laboratory coordinates (real space and time) is conjectured to be equal to the real part of the pole of the exact solution, if the spatial coordinate is allowed to be complex. This feature can be verified readily for the Peregrine breather (lowest order rogue wave) of the nonlinear Schrödinger equation. This connection is further demonstrated numerically here for more complicated scenarios, namely the second order rogue wave of the Boussinesq equation (for bidirectional long waves in shallow water), an asymmetric second order rogue wave for the nonlinear Schrödinger equation (as evolution system for slowly varying wave packets), and a symmetric second order rogue wave of coupled Schrödinger systems. Furthermore, the maximum displacements in physical space occur at a time instant where the trajectories of the poles in the complex plane reverse directions. This property is conjectured to hold for many other systems, and will help to determine the maximum amplitudes of rogue waves.

Item Type: Article
DOI/Identification number: 10.1063/1.5001007
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Peter A Clarkson
Date Deposited: 17 Oct 2017 08:42 UTC
Last Modified: 29 May 2019 19:43 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/64054 (The current URI for this page, for reference purposes)
Clarkson, Peter: https://orcid.org/0000-0002-8777-5284
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