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## Коротковолновые поперечные неустойчивости плоских солитонов в двумерном гиперболическом нелинейном уравнении Шредингера

Theoretical and Mathematical Physics. 2014. Т. 179. № 1. С. 78-89.

Пелиновский Д. Е., Rouvinskaya E., Kurkina O. E., Деконинк Б.

We prove that line solitons of the two-dimensional hyperbolic nonlinear Schr¨odinger equation are unstable
under transverse perturbations of arbitrarily small periods, i.e., short waves. The analysis is based on
the construction of Jost functions for the continuous spectrum of Schr¨odinger operators, the Sommerfeld
radiation conditions, and the Lyapunov–Schmidt decomposition. We derive precise asymptotic expressions
for the instability growth rate in the limit of short periods.

Slunyaev A., Studies in Applied Mathematics 2019 Vol. 142 P. 385-413

Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large
amplitudes tends to other long-wave models, the classic and the modified Korteweg–de Vries equations. The local
solution for an isolated soliton or breather within the GE is obtained. ...

Added: March 11, 2019

Pelinovsky E., Dutykh D., Physical Letters A 2014 Vol. 378 No. 42 P. 3102-3110

The collective behaviour of soliton ensembles (i.e. the solitonic gas) is studied using the methods of the direct numerical simulation. Traditionally this problem was addressed in the context of integrable models such as the celebrated KdV equation. We extend this analysis to non-integrable KdV–BBM type models. Some high resolution numerical results are presented in both ...

Added: November 19, 2014

Pelinovsky E., Touboil J., European J Mechanics B Fluids (Elsivier) 2014 Vol. 48 P. 13-18

The bottom pressure distribution under solitonic waves, travelling or fully reflected at a wall is analysed here. Results given by two kind of numerical models are compared. One of the models is based on the Green–Naghdi equations, while the other one is based on the fully nonlinear potential equations. The two models differ through the ...

Added: November 19, 2014

Pelinovsky D., Slunyaev A., Kokorina A. et al., Communications in Nonlinear Science and Numerical Simulation 2021 Vol. 101 Article 105855

Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to ...

Added: May 11, 2021

Didenkulova I., Nikolkina I., Pelinovsky E., JETP Letters 2013 Vol. 97 No. 4 P. 221-225

Properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20m and ...

Added: February 26, 2013

Boiti M., Pempinelli F., Pogrebkov A., Journal of Mathematical Physics 2011 Vol. 52 No. 083506 P. 1-21

Properties of Jost and dual Jost solutions of the heat equation, F (x,k)
and Y(x,k), in the case of a pure solitonic potential are studied in
detail.We describe their analytical properties on the spectral parameter k
and their asymptotic behavior on the x-plane and we show that the values
of e(−qx)F (x, k) and the residues of exp(qx ...

Added: February 16, 2013

Dymov A. V., Kuksin S., Communications in Mathematical Physics 2021 Vol. 382 P. 951-1014

We consider the damped/driven (modified) cubic NLS equation on a large
torus with a properly scaled forcing and dissipation, and decompose its solutions to
formal series in the amplitude. We study the second order truncation of this series and
prove that when the amplitude goes to zero and the torus’ size goes to infinity the energy
spectrum of the ...

Added: June 29, 2021

Бойти М., Пемпинелли Ф., Pogrebkov A., Теоретическая и математическая физика 2012 Т. 172 № 2 С. 181-197

Рассмотрен оператор теплопроводности с общим многосолитонным потенциалом, выведена его расширенная резольвента, зависящая от параметра . Показана ее ограниченность по всем переменным и разрывность по параметру . Введены функции Грина и детально исследованы их свойства ...

Added: February 18, 2013

Dymov A. V., Kuksin S., Journal of Statistical Physics 2023 Vol. 190 No. 1 Article 3

In this paper we continue to study small amplitude solutions of the damped cubic NLS equation, driven by a random force [the study was initiated in our previous work Dymov and Kuksin (Commun Math Phys 382:951–1014, 2021) and continued in Dymov et al. (The large-period limit for equations of discrete turbulence 2021, arXiv:2104.11967)]. We write solutions ...

Added: October 31, 2022

Chabchoub A., Slunyaev A., Hoffmann N. et al., Frontiers in Physics 2021 Vol. 9 Article 633549

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate ...

Added: August 25, 2021

Dymov A. V., Куксин С. Б., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2020 Т. 491 № 1 С. 29-37

Авторы обсуждают ряд строгих результатов в стохастической модели волновой турбулентности Захарова–Львова. А именно, рассматривают уравнение Шрёдингера с (модифицированной) кубической нелинейностью и вязкостью на торе большого периода, возмущенное случайной силой, и раскладывают его решение в формальный ряд по амплитуде. Авторы показывают, что в пределе, когда
амплитуда стремится к нулю, а период тора – к бесконечности, спектр энергии ...

Added: June 29, 2021

Kalinin N., Shkolnikov M., Communications in Mathematical Physics 2020 No. 378 P. 1649-1675

Let F: Z^2→Z be the pointwise minimum of several linear functions. The theory of smoothing allows us to prove that under certain conditions there exists the pointwise minimal function among all integer-valued superharmonic functions coinciding with F “at infinity”. We develop such a theory to prove existence of so-called solitons (or strings) in a sandpile model, studied by S. Caracciolo, G. Paoletti, and ...

Added: August 25, 2020

Kalinin N., Frontiers in Physics 2020 Vol. 8 Article 581126

Sandpile model exhibits fascinating pattern structures: patches, characterized by quadratic functions, and line-shaped patterns (also called solitons, webs, or linear defects). It was predicted by Dhar and Sadhu that sandpile patterns with line-like features may be described in terms of tropical geometry. We explain the main ideas and technical tools -- tropical geometry and discrete ...

Added: October 29, 2020

Abrashkin A. A., Pelinovsky E., Nonlinear Processes in Geophysics 2017 Vol. 24 P. 255-264

The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift ...

Added: June 26, 2017

Slunyaev A., Кокорина А. В., Journal of Ocean Engineering and Marine Energy 2017 Vol. 3 P. 395-408

The results of the probabilistic analysis of the direct numerical simulations of irregular unidirectional deep
water waves are discussed. It is shown that an occurrence of large-amplitude soliton-like groups represents an extraordinary case, which is able to increase noticeably the probability of high waves even in moderately rough sea conditions. The ensemble of wave realizations should be large enough to take these ...

Added: March 1, 2019

Abrashkin A. A., Bodunova Y., Fluid Dynamics 2013 Vol. 48 No. 2 P. 223-231

Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the ...

Added: February 25, 2014

Abrashkin A. A., Pelinovsky E., Известия РАН. Физика атмосферы и океана 2018 № 1

The nonlinear Schrödinger (NLS) equation describing the propagation of inhomogeneous vertical wave packets in an infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is shown that the modulation instability criteria of the considered ...

Added: October 16, 2017

Kamchatnov A.M., Chaos 2019 Vol. 29 Article 023106

We discuss the problem of breaking of a nonlinear wave in the process of its propagation into a medium at rest. It is supposed that the profile of the wave is described at the breaking moment by the function (−x) 1/n (x < 0, positive pulse) or −x 1/n (x > 0, negative pulse) of ...

Added: February 4, 2021

Gromov Evgeny, Malomed B., Chaos 2016 Vol. 26 No. 12 P. 123118-1-123118-10

One-parameter families of exact two-component solitary-wave solutions for interacting high-frequency (HF) and low-frequency (LF) waves are found in the framework of Zakharov-type models, which couple the nonlinear Schrödinger equation (NLSE) for intense HF waves to the Boussinesq (Bq) or Korteweg - de Vries (KdV) equation for the LF component through quadratic terms. The systems apply, ...

Added: November 26, 2016

Ivanov S. K., Kamchatnov A.M., Physics of Fluids 2019 Vol. 31 Article 057102

We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated, and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage ...

Added: February 4, 2021

O.E. Kurkina, A.A. Kurkin, T. Soomere et al., Physics of Fluids 2011 Vol. 23 No. 11 P. 116602-1-13-116602-13

We address a specific but possible situation in natural water bodies when the three-layer stratification has a symmetric nature, with equal depths of the uppermost and the lowermost layers. In such case, the coefficients at the leading nonlinear terms of the modified Korteweg-de Vries (mKdV) equation vanish simultaneously. It is shown that in such cases ...

Added: November 6, 2012

Abrashkin A. A., Pelinovsky E., Izvestia, Atmospheric and Oceanic Physic 2018 Vol. 54 No. 1 P. 101-105

A nonlinear Schrцdinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously
vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function
of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is
shown that the modulational instability criteria for the weakly vortical waves and ...

Added: October 3, 2018

Slunyaev A., Тарасова Т. В., Chaos 2022 Vol. 32 Article 101102

Synchronous collisions between a large number of solitons are considered in the context of a statistical description. It is shown that, during the interaction of solitons of the same signs, the wave field is effectively smoothed out. When the number of solitons increases and the sequence of their amplitudes decay slower, the focused wave becomes even smoother ...

Added: October 14, 2022

Kurkina O. E., Kurkin A. A., Rouvinskaya E. et al., Письма в Журнал экспериментальной и теоретической физики 2012 Т. 95 № 2 С. 98-103

Nonlinear wave dynamics is discussed using the extended modified Korteweg–de Vries equation that includes the combination of the third- and fifth- order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are ...

Added: August 24, 2012