scholarly journals Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1445
Author(s):  
Ekaterina Didenkulova ◽  
Efim Pelinovsky
Keyword(s):  
Optical Fibers ◽  
Water Waves ◽  
Relative Phase ◽  
Vries Equation ◽  
Decisive Role ◽  
Pairwise Interactions ◽  
De Vries ◽  
Series Of Experiments ◽  
Stratified Ocean ◽  
The Moment

Pairwise interactions of particle-like waves (such as solitons and breathers) are important elementary processes that play a key role in the formation of the rarefied soliton gas statistics. Such waves appear in different physical systems such as deep water, shallow water waves, internal waves in the stratified ocean, and optical fibers. We study the features of different regimes of collisions between a soliton and a breather in the framework of the focusing modified Korteweg–de Vries equation, where cubic nonlinearity is essential. The relative phase of these structures is an important parameter determining the dynamics of soliton–breather collisions. Two series of experiments with different values of the breather’s and soliton’s relative phases were conducted. The waves’ amplitudes resulting from the interaction of coherent structures depending on their relative phase at the moment of collision were analyzed. Wave field moments, which play a decisive role in the statistics of soliton gases, were determined.

The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1

1973 ◽  
Vol 59 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Harvey Segur
Keyword(s):  
Water Waves ◽  
Large Time ◽  
Vries Equation ◽  
Asymptotic Properties ◽  
Series Representation ◽  
Convergent Series ◽  
Asymptotic Results ◽  
De Vries ◽  
Method Of Solution

The method of solution of the Korteweg–de Vries equation outlined by Gardneret al.(1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

Irrotational water waves and the complex Korteweg-de Vries Equation

1996 ◽  
Vol 98 (2-4) ◽  
pp. 510-514 ◽  
Author(s):  
D. Levi ◽  
M. Sanielevici
Keyword(s):  
Water Waves ◽  
Vries Equation ◽  
De Vries

Soliton turbulence in weakly nonlinear and weakly dispersive media

2021 ◽  
Author(s):  
Ekaterina Didenkulova
Keyword(s):  
Water Waves ◽  
Solid Mechanics ◽  
Vries Equation ◽  
Turbulence Theory ◽  
Dispersive Media ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
De Vries ◽  
Interaction Of Solitons ◽  
Soliton Collisions

<p>A short review on weakly nonlinear and weakly dispersive dynamics of soliton ensembles, the so-called soliton turbulence is given. Such processes take place in shallow water waves, internal waves in the atmosphere and the ocean, solid mechanics and astrophysical plasma; they are described by the integrable models of Korteweg – de Vries equation type (modified Korteweg – de Vries equation, Gardner equation). Here, soliton turbulence means an ensemble of solitons with random parameters. The property of solitons to interact elastically with each other gives rise to an obvious association with the gas of elastically colliding particles. Strictly speaking, soliton turbulence (soliton gas) is a deterministic dynamical system due to the integrability of equations describing the evolution of waves (solitons). However, due to the great complexity of its behavior (due to the large number of participating solitons and nonlinear nature of their interactions), the dynamics of the system can be considered random and, accordingly, may be investigated using methods typical for such problems.</p><p>Firstly, pair soliton collisions have been analyzed as an elementary act of the soliton turbulence for further understanding of their impact on multi-soliton dynamics. Different types of solitons have been considered: “thick” or “top-table” solitons, algebraic solitons, solitons of different polarities. From the point of view of the turbulence theory, the interactions of waves (particles) should be described by the statistical moments of the wave field. It was shown that the interaction of solitons of the same polarity leads to a decrease in the third and fourth moments characterizing the skewness and kurtosis. However, the interaction of solitons of different polarity leads to an increase in these moments of the soliton field. Then, the study of collision patterns of breathers (localized oscillating packets) with each other and with solitons has been carried out. The determination of conditions leading to an extreme scenario, as well as statistical properties, probability and features of large wave manifestation has been provided. As a result of numerical modeling of the multi-soliton field dynamics, the appearance of anomalously large waves in bipolar soliton fields has been demonstrated. Though most of the soliton collisions occur between the pairs of solitons, which may result in maximum two-fold wave amplification, multiple collisions also happen (they make about 10% of the total number of collisions). The long-term simulation of the soliton gas dynamics has shown a significant decrease in skewness and significant increase in kurtosis, confirming the fact of abnormally large wave (so-called “freak/rogue wave”) occurrence.</p><p>The reported study was funded by RFBR according to the research projects 19-35-60022 and 21-55-15008.</p>

Soliton turbulence in weakly nonlinear and weakly dispersive media

2020 ◽  
Author(s):  
Ekaterina Didenkulova
Keyword(s):  
Water Waves ◽  
Solid Mechanics ◽  
Vries Equation ◽  
Turbulence Theory ◽  
Dispersive Media ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
De Vries ◽  
Interaction Of Solitons ◽  
Soliton Collisions

<p>A short review on weakly nonlinear and weakly dispersive dynamics of soliton ensembles, the so-called soliton turbulence is given. Such processes take place in shallow water waves, internal waves in the atmosphere and the ocean, solid mechanics and astrophysical plasma; they are described by the integrable models of Korteweg – de Vries equation type (modified Korteweg – de Vries equation, Gardner equation). Here, soliton turbulence means an ensemble of solitons with random parameters. The property of solitons to interact elastically with each other gives rise to an obvious association with the gas of elastically colliding particles. Strictly speaking, soliton turbulence (soliton gas) is a deterministic dynamical system due to the integrability of equations describing the evolution of waves (solitons). However, due to the great complexity of its behavior (due to the large number of participating solitons and nonlinear nature of their interactions), the dynamics of the system can be considered random and, accordingly, may be investigated using methods typical for such problems. </p><p>Firstly, pair soliton collisions have been analyzed as an elementary act of the soliton turbulence for further understanding of their impact on multi-soliton dynamics. Different types of solitons have been considered: “thick” or “top-table” solitons, algebraic solitons, solitons of different polarities. From the point of view of the turbulence theory, the interactions of waves (particles) should be described by the statistical moments of the wave field. These moments, with the exception of the first two, are not invariants of the equation and are not preserved within the time. It was shown that the interaction of solitons of the same polarity leads to a decrease in the third and fourth moments characterizing the skewness and kurtosis. However, the interaction of solitons of different polarity leads to an increase in these moments of the soliton field.</p><p> Then, the study of collision patterns of breathers (localized oscillating packets) with each other and with solitons has been carried out. The determination of conditions leading to an extreme scenario, as well as statistical properties, probability and features of large wave manifestation has been provided. </p><p>As a result of numerical modeling of the multi-soliton fields’ dynamics, the appearance of anomalously large waves in bipolar soliton fields has been demonstrated. Though most of the soliton collisions occur between the pairs of solitons, which may result in maximum two-fold wave amplification, multiple collisions also happen (they make about 10% of the total number of collisions).  The long-term simulation of the soliton gas dynamics has shown a significant decrease in skewness and significant increase in kurtosis, confirming the fact of abnormally large waves’ (so-called “freak/rogue waves”) occurrence.</p><p>The reported study was funded by RFBR according to the research projects 19-35-60022 and 18-02-00042.</p>

scholarly journals PAIRWISE INTERACTIONS OF COHERENT STRUCTURES ON THE SURFACE OF DEEP WATER

2019 ◽  
Vol 47 (1) ◽  
pp. 66-68
Author(s):  
D.I. Kachulin ◽  
A.A. Gelash ◽  
A.I. Dyachenko ◽  
V.E. Zakharov
Keyword(s):  
Wave Field ◽  
Deep Water ◽  
Coherent Structures ◽  
Relative Phase ◽  
Field Amplitude ◽  
Energy Losses ◽  
Exact Model ◽  
Pairwise Interactions ◽  
The Moment ◽  
Maximum Amplification

The interactions of coherent structures (different types of solitary wave groups) on the surface of deep water is an important nonlinear wave process, which has been studied both theoretically and experimentally (Dyachenko et al., 2013a, b; Slunyaev et al., 2017). At the moment, a complete theoretical description of such interactions is known only for the simplest model – the nonlinear Schrödinger equation (NSE) where exact multi-soliton solutions are found. In the work (Kachulin, Gelash, 2018), the dynamics of pairwise interactions of coherent structures (breathers) on the surface of deep water were numerically investigated in the framework of the Dyachenko-Zakharov model. Significant differences were found in the collision dynamics of breathers of the compact Dyachenko-Zakharov equation when compared to the behavior of the NLSE solitons. It was found that in a more precise model of gravitational surface waves, in contrast to the NLSE, the maximum amplification of the wave field amplitude during the collision process of coherent structures can exceed the sum of the initial amplitudes of the breathers. In addition, the maximum amplification of the wave field amplitude increases with increasing steepness of the interacting breathers and exceeds unity by 20% at the value of the wave steepness m ≈ 0.2. It was revealed that an important parameter determining the dynamics of pairwise collisions of breathers is the relative phase of these objects at the moment of interaction. The interaction of breathers in the non-integrable Dyachenko-Zakharov model leads to the appearance of small radiation, which was discovered earlier in 2013 (Dyachenko et al., 2013a, b). In the work (Kachulin, Gelash, 2018) we demonstrate that the magnitude of the energy losses of the colliding solitons to radiation also depends on their relative phase. Maximum of the energy losses is observed at the same relative phase, at which the amplitude amplification maximum is observed. In addition, depending on the value of the relative phase, solitons can both gain and lose the energy, which results in increase or decrease of their amplitude after a collision. It was found that, in contrast to the NSE model, the spatial shifts of solitons in a more precise model can be both positive and negative. We use the exact breather solutions of the Dyachenko-Zakharov model and the canonical transformation to physical variables (the free surface profile and the potential on the liquid surface) to find approximate solutions in the form of breathers within the framework of exact nonlinear equations for potential incompressible fluid flows. The preliminary results of our numerical experiments in the exact model demonstrate similar dynamics of the interaction of breathers, which indicates that the theoretical picture of the interaction of coherent structures presented here is universal and can be observed in laboratory experiments. The study of the dynamics of breather interactions in the exact model performed by D.I. Kachulin was supported by the Russian Science Foundation (Grant No. 18-71-00079). The work of V.E. Zakharov and A.I. Dyachenko on the dynamics of breather interactions in approximate models was supported by the state assignment “Dynamics of the complex materials”.

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