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Fluids 2016, 1(3), 23; doi:10.3390/fluids1030023

The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time

1
Graduate School of Frontier Sciences, The University of Tokyo, Chiba 277-8563, Japan
2
Department of Mechanical Engineering, Aalto University, Espoo 02150, Finland
3
Department of Mathematics, University College London, London, WC1E 6BT, UK
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Pavel Berloff
Received: 11 April 2016 / Revised: 4 July 2016 / Accepted: 11 July 2016 / Published: 19 July 2016
(This article belongs to the Collection Geophysical Fluid Dynamics)
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Abstract

The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability, the NLS provides families of exact solutions describing the dynamics of localised structures which can be observed experimentally in applicable nonlinear and dispersive media of interest. Depending on the co-ordinate of wave propagation, it is known that the NLS can be either expressed as a space- or time-evolution equation. Here, we discuss and examine in detail the limitation of the first-order asymptotic equivalence between these forms of the water wave NLS. In particular, we show that the the equivalence fails for specific periodic solutions. We will also emphasise the impact of the studies on application in geophysics and ocean engineering. We expect the results to stimulate similar studies for higher-order weakly nonlinear evolution equations and motivate numerical as well as experimental studies in nonlinear dispersive media. View Full-Text
Keywords: nonlinear waves; localized structures; rogue waves nonlinear waves; localized structures; rogue waves
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Chabchoub, A.; Grimshaw, R.H.J. The Hydrodynamic Nonlinear Schrödinger Equation: Space and Time. Fluids 2016, 1, 23.

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