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## Formal Expansions in Stochastic Model for Wave Turbulence 2: Method of Diagram Decomposition

Journal of Statistical Physics. 2023. Vol. 190. No. 1. Article 3.

Dymov A. V., Kuksin S.

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 of the equation as formal series in the amplitude and discuss the behaviour of this series under the wave turbulence limit, when the amplitude goes to zero, while the space-period goes to infinity.

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

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

Пелиновский Д. Е., Rouvinskaya E., Kurkina O. E. et al., Теоретическая и математическая физика 2014 Т. 179 № 1 С. 78-89

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 ...

Added: May 13, 2014

Slunyaev A., Степанянц Ю. А., Physics of Fluids 2022 Vol. 34 No. 7 Article 077121

We study the nonlinear modulation property of flexural-gravity waves on a water surface covered by a compressed ice sheet of given thickness and density in a basin of a constant depth. For weakly nonlinear perturbations, we derive the nonlinear Schrodinger equation and investigate the conditions when a quasi-sinusoidal wave becomes unstable with respect to amplitude modulation. The ...

Added: October 13, 2022

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

Kolokoltsov V., Switzerland : Birkhauser/Springer, 2019

This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial ...

Added: November 14, 2019

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

Dymov A. V., Kuksin S., Maiocchi A. et al., The large-period limit for equations of discrete turbulence / Cornell University. Series arXiv "math". 2021.

We consider the damped/driven cubic NLS equation on the torus of a large period L with a small nonlinearity of size λ, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first λ → 0 and then L → ∞. The first limit, called the limit of discrete ...

Added: June 29, 2021

Gromov E.M., Malomed B. A., Physica Scripta 2015 Vol. 90 No. 6 P. 068021-1-068021-6

Dynamics of Langmuir solitons is considered in plasmas with spatially inhomogeneous electron temperature. An underlying Zakharov-type system of two unidirectional equations for the Langmuir and ion-sound fields is reduced to an inhomogeneous nonlinear Schrödinger equation (NLSE) with spatial variation of the second-order dispersion (SOD) and self-phase modulation (SPM) coefficients, induced by the spatially inhomogeneous profile ...

Added: April 25, 2015

Abrashkin A. A., Бодунова Ю. П., Известия РАН. Механика жидкости и газа 2013 № 2 С. 81-89

В рамках лагранжевого подхода разработан метод описания волнового пакета на поверхности бесконечно глубокой вязкой жидкости. Проанализирован случай, когда обратное число Рейнольдса порядка квадрата крутизны волны. Выражения для траекторий жидких частиц определены с точностью до куба крутизны. Указаны условия, при которых эволюция огибающей пакета описывается нелинейным уравнением Шредингера с линейным по амплитуде диссипативным членом. Сформулировано правило, ...

Added: November 19, 2013

Abrashkin A. A., Pelinovsky E., Успехи физических наук 2018 Т. 188 С. 329-334

It is shown in the quadratic approximation that the Gerstner wave vorticity is equal and of different sign to the vorticity of Stokes drift current in deep water. That gives an opportunity to interpret the Stokes wave as a superposition of the Gerstner wave and the Stokes drift and to explain, why the nonlinearity in ...

Added: October 17, 2017

Abrashkin A. A., Pelinovsky E., , in : EGU General Assembly 2016, Geophysical Research Abstracts. : Wien : CC Attribution 3.0 License, 2016.

Nonlinear Schrodinger equation is derived for weakly modulated nonlinear wave packets in the infinite-depth water flow with inhomogeneous vorticity. Governing 2-D equations are written in Lagrangian variables. Nonlinear Schrodinger equation is obtained in the third order of perturbation theory taking into account weak non-uniform vortex current. Two limiting cases are analyzed. The first one corresponds ...

Added: September 27, 2016

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

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

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

Agalarov A. M., Gadzhimuradov T. A., Potapov A. A. et al., Modeling and Analysis of Information Systems 2018 Vol. 25 No. 1 P. 133-139

The multi-component extension problem of the (2 + 1)D-gauge topological Jackiw–Pi model describing the nonlinear quantum dynamics of charged particles in multi-layer Hall systems is considered. By applying the dimensional reduction (2 + 1)D → (1 + 1)D to Lagrangians with the Chern–Simons topologic fields , multi-component nonlinear Schrodinger equations for particles are constructed with ...

Added: December 20, 2022

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

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

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

Brazhnikov M., Левченко А. А., Межов-Деглин Л. П. et al., Письма в Журнал экспериментальной и теоретической физики 2015 Т. 100 № 10 С. 753-759

Formation of low frequency harmonics on turbulent distribution in the system of waves on the surface of liquid hydrogen has been studied in the frequency range 1-100 Hz (capillary-gravity waves). It is shown that the geometry of the experimental cell has a significant influence on the direct cascade of capillary waves generated by monochromatic force ...

Added: January 20, 2019

Abrashkin A. A., Pelinovsky E., Physics-Uspekhi 2018 Vol. 61 P. 307-312

We discuss the properties of two-dimensional, non
linear, potential, and vortex waves on the surface of an ideal
liquid of infinite depth. It is shown that in the quadratic order in
the amplitude, the vorticity of the Gerstner wave is equal in
magnitude to and different in sign from that of the Stokes drift
current in a surface layer. This ...

Added: October 3, 2018

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Litvin Y. V., Абрамов И. В., Технологии техносферной безопасности 2016 № 66

Advanced approach to the assessment of a random time of arrival fire fighting calculation on the object of protection, the time of their employment and the free combustion. There is some quantitative assessments with the review of analytical methods and simulation ...

Added: August 27, 2016