Article information

2019 , Volume 24, ¹ 2, p.52-66

Didenkulova E.G., Kokorina A.V., Slyunyaev A.Y.

Numerical simulation of soliton gas within the Korteweg-de Vries type equations

The details of the numerical scheme and the method of specifying the initial conditions for the simulation of the irregular dynamics of soliton ensembles within the framework of equations of the Korteweg - de Vries type are given using the example of the modified Korteweg - de Vries equation with a focusing type of nonlinearity. The numerical algorithm is based on a pseudo-spectral method with implicit integration over time and uses the Crank-Nicholson scheme for improving the stability property. The aims of the research are to determine the relationship between the spectral composition of the waves (the Fourier spectrum or the spectrum of the associated scattering problem) and their probabilistic properties, to describe transient processes and the equilibrium states.

The paper gives a qualitative description of the evolution of statistical characteristics for ensembles of solitons of the same and different polarities, obtained as a result of numerical simulations; the probability distributions for wave amplitudes are also provided. The results of test experiments on the collision of a large number of solitons are discussed: the choice of optimal conditions and the manifestation of numerical artifacts caused by insufficient accuracy of the discretization. The numerical scheme used turned out to be extremely suitable for the class of the problems studied, since it ensures good accuracy in describing collisions of solitons with a short computation time.

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Keywords: soliton gas, Korteweg-de Vries equation, modified Korteweg - de Vries equation, Gardner equation, numerical simulation

doi: 10.25743/ICT.2019.24.2.005

Author(s):
Didenkulova Ekaterina Gennadievna
PhD.
Position: Research Scientist
Office: Institute of Applied Physics RAS, Nizhny Novgorod State Technical University n.a. R.E. Alekseev
Address: 603950, Russia, Nizhny novgorod, 46 Ulyanova Str.
Phone Office: (831) 4164749
E-mail: eshurgalina@mail.ru
SPIN-code: 3193-2111

Kokorina Anna Vitalievna
PhD.
Position: Research Scientist
Office: Institute of Applied Physics RAS
Address: 603950, Russia, Nizhny novgorod, 46 Ulyanova Str.
Phone Office: (831) 4164749
E-mail: Slunyaev@appl.sci-nnov.ru

Slyunyaev A.Yu.
Office: Institute of Computational Technologies SB RAS
Address: 630090, Russia, Novosibirsk, ac. Lavrentyev Ave. 6
Phone Office: (383) 330 61 68
E-mail: a.slyunyaev@ngs.ru

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Bibliography link:
Didenkulova E.G., Kokorina A.V., Slyunyaev A.Y. Numerical simulation of soliton gas within the Korteweg-de Vries type equations // Computational technologies. 2019. V. 24. ¹ 2. P. 52-66
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