An approximate theoretical description of solitonic gases in the integrable models like the Korteweg-de Vries equation was proposed by V. Zakharov [1]. He has shown that the pair interaction of solitons plays important role in the dynamics of the ensemble of solitary waves. Later this research direction has been successfully pursued by G. El and his co-authors using the inverse scattering technique [2]. In our study we analyse theoretically the pair soliton interaction and numerically the dynamics of random soliton ensembles in integrable (Korteweg-de Vries, modified KdV, Gardner equations) systems and non-integrable (Benjamin-Bona-Maxoni equation) systems.
Firstly, the known two-soliton solutions of the integrable KdV, mKdV and Gardner equations are used to calculate the moments of wave field.
We have shown that the two soliton interaction leads to significant variations of the third and fourth moments in the dominant interaction region while the first two moments are integrals of these equations and remain unchanged due to the conservation of the mass and momentum. In particular, higher moments are decreased if solitons are of the same polarity and increased if solitons have opposite polarities. The magnitudes of the relative variations of the third and fourth moments turn out to be non-monotonic functions of the soliton amplitude ration having a maximum in the transition region between the exchange and overtaking scenarios of two soliton interactions. The qualitative implication of this dynamical effect for the soliton turbulence theory will be a change of the skewness and kurtosis of the turbulent wave field in the regions of higher density of solitons. For the KdV equation these results are published in [3].
Several numerical experiments on the solitonic gas turbulence in the framework of an integrable KdV and a nonintegrable regularized KdV?BBM equation are presented. First of all, we showed that the probability distribution for the solitonic gas remains quasi-invariant during the system evolution for both KdV and KdV?BBM cases. The special attention was paid to the statistical characteristics such as kurtosis and skewness which measure the ?heaviness? of tails and the asymmetry of the free surface elevation distribution. In particular, using the asymptotic methods and Monte?Carlo simulations we showed that both skewness and kurtosis increase with the Stokes?Ursell number S and decrease when the BBM term coefficient. When both parameters are increased gradually and simultaneously, these effects are in competition: first we observe the increase of these statistical characteristics, but then, this tendency is inversed and they decrease after reaching their respective maximal values. We would like to underline that the proposed Monte?Carlo methodology is much less computationally expensive than direct numerical simulations. Despite the small number of Monte?Carlo runs (~ 100) the estimated statistical error is sufficiently small for the purposes of this study. On the other hand, this approach is restricted, strictly speaking, to the situations where the solitons are well separated in space. These results are published in [4].
The present study opens a number of perspectives for future investigations. More general nonlinearities could be included into the model along with some weak dissipative and forcing effects. This could allow us to observe Kolmogorov spectra of a solitonic gas. The nonintegrable effects need some time to be accumulated. Consequently, even longer simulation times are needed. The interaction of a solitonic gas with a random radiation field has to be studied as well.
References:
1) V. E. Zakharov. Kinetic Equation for Solitons. Sov. Phys. - JETP, 60:993?1000, 1971
2) G. El and A. Kamchatnov. Kinetic Equation for a Dense Soliton Gas. Phys. Rev. Lett, 95(20):204101, 2005
3) E. Pelinovsky, E. Shurgalina, A. Sergeeva, T. Talipova, G. El and R. Grimshaw. Two-soliton interaction as an elementary act of soliton turbulence in integrable systems. Physics Letters A, 377: 272-275, 2013.
4) D. Dutykh and E. Pelinovsky. Numerical simulation of a solitonic gas in some integrable and non-integrable models. ArXiv: 1312.165[v]