Method for Solving the Korteweg-deVries Equation

Gardner, Clifford S.; Greene, J. M.; Kruskal, Martin D.; Miura, Robert M.

doi:10.1103/physrevlett.19.1095

Cited by 4,131 publications

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“…Solitons occur in media as diverse as water, DNA, plasma, or ultra-cold gases, 1,[7][8][9][10][11][12] but over the last 20 years optics has led the way in our understanding of soliton interactions because of the ease with which optical solitons can be studied experimentally. [13][14][15][16] Optical solitons have been shown to attract, repel, breakup, merge, orbit each-other or even annihilate.…”

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confidence: 99%

Ultraweak long-range interactions of solitons observed over astronomical distances

et al. 2013

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We report what we believe is the weakest interaction between solitons ever observed. Our experiment involves temporal optical cavity solitons recirculating in a coherently-driven passive optical fibre ring resonator. We observe two solitons, separated by up to 8,000 times their width, changing their temporal separation by a fraction of an attosecond per round-trip of the 100 m-long resonator, or equivalently 1/10,000 of the wavelength of the soliton carrier wave per characteristic dispersive length. The interactions are so weak that, at the speed of light, they require an effective propagation distance of the order of an astronomical unit to fully develop, i.e. tens of millions of kilometres. The interaction is mediated by transverse acoustic waves generated in the optical fibre by the propagating solitons through electrostriction.Solitons are self-localized wave packets that do not spread, the dispersion of the supporting medium being cancelled by a nonlinear effect. 1-4 They are universal, and in many respects behave like particles. 2 They exert forces on each other and can interact in various ways, elastically or inelastically. 2,5,6 Here we report what we believe is by far the weakest form of soliton interaction ever observed. Using recirculating optical cavity solitons, we report interactions so weak that the solitons shift their positions by only about 10 −7 of their width -amounting to 1/10,000 of the wavelength of the soliton carrier wave -per characteristic dispersive length. At the speed of light, these interactions require effective propagation distances of the order of an astronomical unit (AU) to be revealed, i.e. tens of millions of kilometres. The sheer fact that we can actually observe such ultra-weak interactions in a noisy laboratory environment highlights the robustness and stability of solitons as never-before.Solitons occur in media as diverse as water, DNA, plasma, or ultra-cold gases, 1,7-12 but over the last 20 years optics has led the way in our understanding of soliton interactions because of the ease with which optical solitons can be studied experimentally. [13][14][15][16] Optical solitons have been shown to attract, repel, breakup, merge, orbit each-other or even annihilate. 5,17-21 As in various other media, soliton interactions can be either shortrange, occurring when the tails of neighbouring solitons overlap, 6,17,22,23 or long range, through a coupling with a non-local response, be it optical radiation, charge carriers, or thermal waves. 24-27 The weakest soliton interactions reported are long range, 24,25 but their observation is typically limited by the duration or propagation length over which the solitons can be maintained. In optics, this is often simply dictated by the size of a nonlinear crystal or the length of an optical fibre. To overcome this restriction, our experiment involves solitons recirculating in an optical fibre loop.More specifically, we consider temporal cavity solitons (CSs) propagating in a coherently-driven passive nonlinear fibre ring resonator. 28 Not...

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Ultraweak long-range interactions of solitons observed over astronomical distances

et al. 2013

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“…Обратное преобразование рассеяния (ОПР) было введено Гарднером, Грином, Крускалом и Миурой [1] применительно к уравнению Кортевега-де Фриза (КдФ); еще один пример подобного подхода возник в классической работе Захарова и Шаба-та [2] о нелинейном уравнении Шредингера (НУШ). Вскоре после этого Вадати [3] показал, что в ту же схему укладывается и модифицированное уравнение КдФ.…”

Section: Introductionunclassified

Интегрируемые Системы И Квадраты Собственных Функций

Кауп¹,

Kaup²

2009

ТМФ

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Д. Дж. Кауп * ИНТЕГРИРУЕМЫЕ СИСТЕМЫ И КВАДРАТЫ СОБСТВЕННЫХ ФУНКЦИЙДан краткий обзор формализма Абловица-Каупа-Ньюэла-Сегуры для ин-тегрируемых (1D+1D)-систем начиная с пары Лакса, а далее рассмотрена ин-тегрируемая теория возмущений и квадраты собственных функций. Основное внимание уделяется общим свойствам, обнаруженным в обратном преобразова-нии рассеяния для широкого класса известных (1D+1D)-систем. Предпринят ряд шагов, выбранных таким образом, чтобы продвигаться вперед, как при ана-лизе систем высшего порядка. Приведен краткий обзор прямой задачи рассея-ния и обратных задач рассеяния, вслед за чем рассмотрены возмущения потен-циалов и данных рассеяния. Для последней задачи исходный подход к возмуще-ниям системы Абловица-Каупа-Ньюэла-Сегуры переформулирован так, чтобы подчеркнуть его единонаправленность с общими свойствами (1D+1D)-систем. Для вывода возмущений потенциалов, вызванных возмущениями данных рас-сеяния в отсутствие солитонов, использован недавно разработанный подход. Наконец, показано, что последние результаты по нахождению квадратов соб-ственных функций и их сопряженных в виде сумм произведений (а не просто произведений) функций Йоста определяются симметриями, наложенными на матрицу потенциалов.Ключевые слова: солитон, обратное преобразование рассеяния, возмущение, квадрат собственной функции, замыкание, скалярное произведение.

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“…The inverse scattering method (ISM) was introduced first for the Korteweg-de Vries equation (KdVE) [24]. Later it was extended by Zakharov and Shabat [25] to a 2 × 2 scattering problem for the nonlinear Schrödinger equation (NLSE) and that was subsequently generalized by Ablowitz, Kaup, Newell and Segur (AKNS) [26] to include a variety of NLEEs.…”

Section: Exact Solutions For Ibragimov-shabat Equationmentioning

confidence: 99%