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      Rogue waves in the two dimensional nonlocal nonlinear Schrödinger equation and nonlocal Klein-Gordon equation

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      PLoS ONE

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          Abstract

          In this paper, we investigate two types of nonlocal soliton equations with the parity-time ( PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota’s bilinear method. Like the nonlocal NLS equation, these solutions may have singularities. However, by suitable constraints of parameters, nonsingular breather solutions are generated. Besides, by taking a long wave limit of these obtained soliton solutions, rogue wave solutions and semi-rational solutions are derived. For the two dimensional NLS equation, rogue wave solutions are line rogue waves, which arise from a constant background with a line profile and then disappear into the same background. The semi-rational solutions shows intriguing dynamical behaviours: line rogue wave and line breather arise from a constant background together and then disappear into the constant background again uniformly. For the coupled nonlocal Klein-Gordon equation, rogue waves are localized in both space and time, semi-rational solutions are composed of rogue waves, breathers and periodic line waves. These solutions are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides.

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          Most cited references 74

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          Observation of parity–time symmetry in optics

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            Parity-time synthetic photonic lattices.

            The development of new artificial structures and materials is today one of the major research challenges in optics. In most studies so far, the design of such structures has been based on the judicious manipulation of their refractive index properties. Recently, the prospect of simultaneously using gain and loss was suggested as a new way of achieving optical behaviour that is at present unattainable with standard arrangements. What facilitated these quests is the recently developed notion of 'parity-time symmetry' in optical systems, which allows a controlled interplay between gain and loss. Here we report the experimental observation of light transport in large-scale temporal lattices that are parity-time symmetric. In addition, we demonstrate that periodic structures respecting this symmetry can act as unidirectional invisible media when operated near their exceptional points. Our experimental results represent a step in the application of concepts from parity-time symmetry to a new generation of multifunctional optical devices and networks.
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              Optical rogue waves.

              Recent observations show that the probability of encountering an extremely large rogue wave in the open ocean is much larger than expected from ordinary wave-amplitude statistics. Although considerable effort has been directed towards understanding the physics behind these mysterious and potentially destructive events, the complete picture remains uncertain. Furthermore, rogue waves have not yet been observed in other physical systems. Here, we introduce the concept of optical rogue waves, a counterpart of the infamous rare water waves. Using a new real-time detection technique, we study a system that exposes extremely steep, large waves as rare outcomes from an almost identically prepared initial population of waves. Specifically, we report the observation of rogue waves in an optical system, based on a microstructured optical fibre, near the threshold of soliton-fission supercontinuum generation--a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input. We model the generation of these rogue waves using the generalized nonlinear Schrödinger equation and demonstrate that they arise infrequently from initially smooth pulses owing to power transfer seeded by a small noise perturbation.
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                Author and article information

                Contributors
                Role: ConceptualizationRole: MethodologyRole: Writing – review & editing
                Role: Formal analysis
                Role: Methodology
                Role: Editor
                Journal
                PLoS One
                PLoS ONE
                plos
                plosone
                PLoS ONE
                Public Library of Science (San Francisco, CA USA )
                1932-6203
                2018
                12 February 2018
                : 13
                : 2
                Affiliations
                [1 ] College of Mathematic and Information Science, Shandong Technology and Business University, Yantai, 264005, P. R. China
                [2 ] Chuo College of Technology, Tokyo, 114-8543, Japan
                China University of Mining and Technology, CHINA
                Author notes

                Competing Interests: The authors have declared that no competing interests exist.

                Article
                PONE-D-17-38868
                10.1371/journal.pone.0192281
                5809061
                29432495
                © 2018 Liu et al

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                Counts
                Figures: 5, Tables: 0, Pages: 16
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                Funding
                The authors received no specific funding for this work.
                Categories
                Research Article
                Earth Sciences
                Marine and Aquatic Sciences
                Oceanography
                Ocean Waves
                Computer and Information Sciences
                Systems Science
                Nonlinear Systems
                Physical Sciences
                Mathematics
                Systems Science
                Nonlinear Systems
                Computer and Information Sciences
                Systems Science
                Nonlinear Dynamics
                Physical Sciences
                Mathematics
                Systems Science
                Nonlinear Dynamics
                Physical Sciences
                Materials Science
                Materials by Structure
                Mixtures
                Physical Sciences
                Physics
                Particle Physics
                Elementary Particle Interactions
                Fundamental Interactions
                Physical Sciences
                Physics
                Quantum Mechanics
                Physical Sciences
                Physics
                Quantum Mechanics
                Quantum Chromodynamics
                Physical Sciences
                Physics
                Waves
                Traveling Waves
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