Soliton turbulence in weakly nonlinear and weakly dispersive media

<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>

Effect of Fourth-Order Dispersion on Solitonic Interactions

Solitons became important in optical communication systems thanks to their robust nature. However, the interaction of solitons is considered as a bad effect. To avoid interactions, the obvious solution is to respect the temporal separation between two adjacent solitons determined as a bit rate. Nevertheless, many better solutions exist to decrease the bit rate error. In this context, the aim of our work is to study the possibility to delete the interaction of adjacent solitons, by using a special dispersion management system, precisely by introducing both of the third- and fourth-order dispersions in the presence of a group velocity dispersion. To study the influence of the fourth- and third-order dispersions, we use the famous non-linear Schr¨odinger equation solved with the Fast Fourier Transform method. The originality of this work is to bring together the dispersion of the fourth, third, and second orders to separate two solitons close enough to create the Kerr-induced interaction and consequently to improve the propagation by decreasing the bit rate error. This study illustrates the influence of the fourth-order dispersion on one single soliton and two co-propagative solitons with different values of the temporal separation. Then the third order dispersion is introduced in the presence of the fourth-order dispersion in the propagation of one and two solitons in order to study its influence on the interaction. Finally, we show the existence of a precise dispersion management system that allows one to avoid the interaction of solitons.

Soliton turbulence in weakly nonlinear and weakly dispersive media

<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>