Numerical simulation of a solitonic gas in KdV and KdV–BBM equations

doi:10.1016/j.physleta.2014.09.008

Physics Letters A

Volume 378, Issue 42, 28 August 2014, Pages 3102-3110

https://doi.org/10.1016/j.physleta.2014.09.008 Get rights and content

Highlights

•
High-resolution simulations of a solitonic gas are presented.
•
Integrable and non-integrable cases are considered.
•
The effect of integrability was shown to be negligible.
•
Dependence of the statistical characteristics on the model parameters was studied.

Abstract

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 integrable and nonintegrable cases. Moreover, the free surface elevation probability distribution is shown to be quasi-stationary. Finally, we employ the asymptotic methods along with the Monte Carlo simulations in order to study quantitatively the dependence of some important statistical characteristics (such as the kurtosis and skewness) on the Stokes–Ursell number (which measures the relative importance of nonlinear effects compared to the dispersion) and also on the magnitude of the BBM term.

Introduction

Solitary wave solutions play the central role in various nonlinear sciences ranging from hydrodynamics to solid and plasma physics [44], [34], [30]. These solutions can propagate without changing its shape. However, the most intriguing part consists in how these solutions interact with each other. The binary interactions of solitary waves have been studied in the context of various nonlinear wave equations [44], [23], [39], [25], [9], [6]. It is well known that in integrable models the collision of two solitons is elastic, i.e. they interact without emitting any radiation. In non-integrable models usually the interactions are nearly elastic [4].

The collective behaviour of soliton ensembles is much less understood nowadays. When a large number of solitary waves are considered simultaneously the researchers usually speak about the so-called solitonic turbulence or a solitonic gas. The literature on this topic is abundant. Some recent studies on solitonic gas turbulence in the KdV framework include Refs. [31], [35], [32]. The solitonic turbulence in nonintegrable NLS-type equations was studied in [47], [12] and the authors showed that in conservative nonintegrable systems the solitonic gas is a statistical attractor whose dimension decreases with time. Recently it was shown both numerically and experimentally that solitonic ensembles appear in the laminar–turbulent transition in a fibre laser [41], modeled by a non-integrable nonlinear Schrödinger-type equation. However, the dominant number of studies is based on integrable models. This apparent contradiction motivated mainly our investigation to quantify the non-integrable effects onto the collective behaviour of solitons.

An approximate theoretical description of solitonic gases was proposed by V. Zakharov (1971) [45] using the kinetic theory. Later this research direction has been successfully pursued by G. El and A. Kamchatnov [14], [15] who used the Inverse Scattering Technique (IST) [1] limited only to the integrable models. In this study the problem of solitonic gases will be investigated using the methods of direct numerical simulation. The evolution of random wave fields including solitonic gases was simulated numerically in [31], [35], [10] using symplectic, multi-symplectic and pseudo-spectral methods. However, previous investigators considered only a limited number of solitons (a few dozens) to simulate a solitonic gas. In this study we will adopt the pseudo-spectral method since it provides the high accuracy and computational efficiency necessary to handle large computational domains. Our goal will consist in:

•
Investigating the influence of soliton interactions on statistical characteristics of the wave field;
•
Constructing the Probability Density Function (PDF) and compute the first four statistical moments of the solitonic turbulence;
•
Studying the role of non-integrable terms on the characteristics of soliton ensembles.

The present manuscript is organized as follows. In Section 2 we derive the governing equation used in this study and in Section 3 numerical results on a solitonic gas dynamics are presented. The main conclusions of this study are outlined in Section 4.

Section snippets

Mathematical model

As the starting point we choose the celebrated Korteweg–de Vries equation [22], [24], [20], [31] (in dimensional variables) which models the undirectional propagation (here in the rightwards direction) of weakly nonlinear and weakly dispersive waves: $η_{t} + c (1 + \frac{3 h}{2} η) η_{x} + \frac{c h^{2}}{6} η_{x x x} = 0,$ where $η (x, t)$ is the vertical excursion of the free surface above the still water level, h is the uniform undisturbed water depth and $c = \sqrt{g h}$ is the speed of linear gravity waves (g being the gravity acceleration).

The KdV

Numerical results

In order to solve numerically equation (2.3) we use a Fourier-type pseudo-spectral method with 3/2-antialiasing rule [40]. For the time discretization we use the Verner's embedded adaptive $9 (8)$ Runge–Kutta scheme [43]. The time step is chosen adaptively using the so-called $H_{211B}$ digital filter [37], [38] to meet some prescribed error tolerance (generally of the order of machine precision $\sim 10^{- 15}$ ). The number of Fourier modes, the length of the computational domain and other numerical parameters

Conclusions and perspectives

In this study we presented several numerical experiments on the solitonic gas turbulence in the framework of an integrable KdV and a nonintegrable regularized KdV–BBM equation. The numerical results reported above generalize previous investigations [31], [10] where only a limited number (a few dozens) of solitons were used to represent a solitonic gas. Consequently, we reduce the statistical error according to the law of large numbers.

First of all, we showed that the probability distribution

Acknowledgements

D. Dutykh acknowledges the support from ERC under the research project ERC-2011-AdG 290562-MULTIWAVE. E. Pelinovsky would like to thank for the support the Russian State contract (2014/133 – theoretical part), the VolkswagenStiftung, RFBR grants (14-05-00092). The authors would like to thank Professors Gennady El and Al Osborne for stimulating discussions on the topics of integrability and solitons theory.

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Citation Excerpt :
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View full text

Numerical simulation of a solitonic gas in KdV and KdV–BBM equations

Highlights

Abstract

Introduction

Section snippets

Mathematical model

Numerical results

Conclusions and perspectives

Acknowledgements

Comput. Phys. Commun.

J. Comput. Phys.

J. Comput. Phys.

Phys. Lett. A

Eur. J. Mech. B, Fluids

Phys. Lett. A

Math. Comput. Simul.

J. Comput. Appl. Math.

Solitons and the Inverse Scattering Transform

Model equations for long waves in nonlinear dispersive systems

Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci.

Wave Mechanics for Ocean Engineering

A model for the two-way propagation of water waves in a channel

Math. Proc. Camb. Philos. Soc.

Head-on collision of two solitary waves and residual falling jet formation

Nonlinear Process. Geophys.

Reflection of a high-amplitude solitary wave at a vertical wall

J. Fluid Mech.

Solitary water wave interactions

Phys. Fluids

Geometric numerical schemes for the KdV equation

Comput. Math. Math. Phys.

Finite volume methods for unidirectional dispersive wave models

Int. J. Numer. Methods Fluids

Soliton turbulence in nonintegrable wave systems

JETP Lett.

Kinetic equation for a dense soliton gas

Phys. Rev. Lett.

Kinetic equation for a soliton gas and its hydrodynamic reductions

J. Nonlinear Sci.

A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions

J. Fluid Mech.

Method for solving the Korteweg–deVries equation

Phys. Rev. Lett.

Random Seas and Design of Maritime Structures