scholarly journals Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

Author(s):  
Marco Bertola ◽  
Gennady A. El ◽  
Alexander Tovbis

Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalized rogue wave notion then naturally enters as a large-amplitude localized coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalized rogue waves.

Author(s):  
Джордж Парарас-Караяннис

Пересмотренные официальные записи Следственного суда ВМС США содержат выводы о том, что разрушение бронированного крейсера USS Memphis 29 августа 1916 года на якоре у гавани Санто-Доминго (Сьюдад-Трухильо) Доминиканской Республики, остров Эспаньола, вероятно, было вызвано «тропическим волнением» «сейсмической бурей» или «цунами». Тем не менее, современный анализ этой морской катастрофы свидетельствует о том, что гибель корабля произошла не по какой-либо из этих причин, а из-за волн-убийц метеоцунами, вызванных быстрым, значительным и прогрессирующим падением атмосферного давления, которое началось в районе около 22 августа и было связано с проходящим ураганом, который в его самой близкой точке был около 250 морских миль на юг. Кроме того, штормовые волны от этого урагана двигались в направлении Санто-Доминго, преломляясь в резонансе у берега, и усиливались и трансформировались низким барометрическим давлением, мелким континентальным шельфом и местными особенностями побережья и батиметрией залива. Настоящий анализ основан на тщательном изучении судового журнала и наблюдений за событиями со стороны экипажа и людей на берегу. Учитывая ограниченные метеорологические данные того периода времени, в настоящем анализе использовался эмпирический подход для грубой оценки функции распределения Рэлея, верхнего предела изменчивости высоты штормовой волны вдали от наиболее интенсивных потоков ветра, а также максимального периода, длины волны и амплитуды генерируемых штормовых волн в глубине . Основываясь на теориях кноидальных волн и волн Эйри, период и скорость наиболее значительных экстремальных внутренних волн имели метеорологическое происхождение, которое было преобразовано в мелкой воде в результате резонансного и наложенного прихода двух других волн, которые создали трехступенчатое плато, на переднем фронте огромная одиночная волна-убийца метеоцунами высотой около 70 футов, с тремя четкими ступенями, двумя плато на передней поверхности и предшествующей впадиной длиной около 300 футов. Основываясь на этом анализе, настоящее исследование пришло к выводу, что именно эта значительная волна метеоцунами / волна-убийца в сочетании с одновременно прибывающими штормовыми волнами охватила Мемфис USS в 16 ч. 40 м. 29 августа 1916 года, разорвав цепи якорей и разрушая его на скалах Санто-Доминго. Official revised records of a U.S. Navy Court of Inquiry concluded that the 29 August 1916 destruction of the armored cruiser USS Memphis anchored off Santo Domingo (Ciudad Trujillo) harbor of the Dominican Republic, Island of Hispaniola, was probably caused by a “tropical disturbance”, a “seismic storm”, or a “tsunami”. However, the present analysis of this naval disaster documents that the loss of the ship was not due to any of these causes, but to rogue waves of a meteotsunami generated from a rapid, significant and progressive drop in atmospheric pressure which begun in the area around August 22 and was associated with a passing hurricane which at its closest point was about 250 nautical miles to the south. Also, storm waves from this hurricane moved towards Santo Domingo refracting in resonance near shore and were further amplified and transformed by the low barometric pressure, the shallow continental shelf and the local coastal features and bathymetry of the bay. The present analysis is based on careful examination of the ship’s log, and on observations of events by the crew and people on the shore. Given the limited meteorological data of that time period, the present analysis used an empirical approach to roughly evaluate the Rayleigh distribution function, the upper limit of storm wave height variability away from the most intense wind fetches, as well as the maximum period, wavelength and deep-water heights of generated storm waves. Based on Airy and cnoidal wave theories, the deep water period and celerity of the most significant extreme wave was of meteorological origin which was transformed in shallower water by the resonant and superimposed arrival of two other waves which created a three step plateau on the face of a huge single rogue wave of the meteotsunami, estimated to be about 70 feet in height, with three distinct steps, two plateaus on its forward face, and a preceding trough estimated to be 300 ft. long. Based on this analysis, the present study concluded that it was this significant meteotsunami/rogue wave, in combination with concurrently arriving storm swells, that engulfed the USS Memphis at 1640 hour in the afternoon of 29 August 1916 - breaking the chains of its anchors and wrecking it on the rocks of Santo Domingo.


2019 ◽  
Vol 33 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.


2020 ◽  
Vol 34 (23) ◽  
pp. 2050234
Author(s):  
Yong Chen ◽  
Xiu-Bin Wang ◽  
Bo Han

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger equation, which is a generalization of the standard nonlinear Schrödinger equation. By means of the modified Darboux transformation, the hierarchies of rational solutions and breather solutions are generated from the plane wave solution. Furthermore, the main characteristics of the nonlinear waves including the Akhmediev breathers, Kuznetsov–Ma solitons, and their combined structures are graphically discussed. Our results would be of much importance in enriching and explaining rogue wave phenomena in nonlinear wave fields.


2016 ◽  
Vol 30 (13) ◽  
pp. 1650208 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Sha-Sha Yuan ◽  
Yue Wang

In this paper, the generalized Darboux transformation for the coherently-coupled nonlinear Schrödinger (CCNLS) system is constructed in terms of determinant representations. Based on the Nth-iterated formula, the vector bright soliton solution and vector rogue wave solution are systematically derived under the nonvanishing background. The general first-order vector rogue wave solution can admit many different fundamental patterns including eye-shaped and four-petaled rogue waves. It is believed that there are many more abundant patterns for high order vector rogue waves in CCNLS system.


Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Sha. Zhou ◽  
Qian Wang

The similarity transformation and direct ansatz are applied to obtain rogue wave solutions of nonlinear Schrödinger equation with varying coefficients. These obtained solutions can be used to describe the possible formation mechanisms for optical rogue wave phenomenon in optical fibres. Moreover their dynamical behaviors are exhibited for chosen different functions. This will further excite the possibility of relative researchers and potential applications of rogue waves in other related fields.


2015 ◽  
Vol 70 (5) ◽  
pp. 365-374 ◽  
Author(s):  
Qi-Min Wang ◽  
Yi-Tian Gao ◽  
Chuan-Qi Su ◽  
Yu-Jia Shen ◽  
Yu-Jie Feng ◽  
...  

AbstractIn this article, a fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort optical pulses, up to the attosecond duration, in an optical fibre. Rogue wave solutions are derived by virtue of the generalised Darboux transformation. Rogue wave structures and interaction are discussed through (i) the analyses on the higher-order rogue waves, the cubic, quartic, quintic, group-velocity, and phase-parameter effects; (ii) a higher-order rogue wave consisting of the first-order rogue waves via the interaction; (iii) characteristics of the rogue waves which are summarised, including the maximum/minimum values of the rogue waves and the number of the first-order rogue waves for composing the higher-order rogue wave; and (iv) spatial-temporal patterns which are illustrated and compared with those of the ‘self-focusing’ nonlinear Schrödinger equation. We find that the quintic terms increase the time of appearance for the first-order rogue waves which form the higher-order rogue wave, and that the quintic terms affect the interaction among the first-order rogue waves, which elongates the distance of appearance for the higher-order rogue wave.


2016 ◽  
Vol 71 (2) ◽  
pp. 95-101 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Xiao-Li Liu ◽  
Li-Li Wen

AbstractIn this paper, a (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which is a generalisation of the NLS equation, is under investigation. The classical and generalised N-fold Darboux transformations are constructed in terms of determinant representations. With the non-vanishing background and iterated formula, a family of the analytical solutions of the (2+1)-dimensional NLS equation are systematically generated, including the bright-line solitons, breathers, and rogue waves. The interaction mechanisms between two bright-line solitons and among three bright-line solitons are both elastic. Several patterns for first-, second, and higher-order rogue wave solutions fixed at space are displayed, namely, the fundamental pattern, triangular pattern, and circular pattern. The two-dimensional space structures of first-, second-, and third-order rogue waves fixed at time are also demonstrated.


2019 ◽  
Vol 33 (08) ◽  
pp. 1950090
Author(s):  
Xiao-Yu Wu ◽  
Bo Tian ◽  
Zhong Du ◽  
Xia-Xia Du

Lattices are used in such fields as electricity, optics and magnetism. Under investigation in this paper is an inhomogeneous discrete nonlinear Schrödinger equation, which models the wave propagation in a lattice. Employing the Kadomtsev–Petviashvili (KP) hierarchy reduction, we obtain the rogue-wave solutions, and see that the rogue waves are affected by the coefficient of the on-site external potential. We see (1) the first-order rogue wave with one peak and two hollows; (2) the second-order rogue waves, each of which is with one peak or three humps; (3) the third-order rogue waves, each of which is with one peak or six humps, and those humps exhibit the triangular pattern, anti-triangular pattern and circular pattern. When the coefficient of the on-site external potential is a constant, the rogue waves periodically appear. When the coefficient of the on-site external potential monotonously changes, oscillations emerge on the constant background.


Sign in / Sign up

Export Citation Format

Share Document