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Zh. Vychisl. Mat. Mat. Fiz., 2019, Volume 59, Number 1, Pages 158–168 (Mi zvmmf10825)  

Compacton solutions of the Korteweg–de Vries equation with constrained nonlinear dispersion

S. P. Popov

Dorodnicyn Computing Center, Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Moscow, 119333 Russia

Abstract: The numerical solution of initial value problems is used to obtain compacton and kovaton solutions of $\mathrm{K}(f^m,g^n)$ equations generalizing the Korteweg–de Vries $\mathrm{K}(u^2,u^1)$ and Rosenau–Hyman $\mathrm{K}(u^m,u^n)$ equations to more general dependences of the nonlinear and dispersion terms on the solution $u$. The functions $f(u)$ and $g(u)$ determining their form can be linear or can have the form of a smoothed step. It is shown that peakocompacton and peakosoliton solutions exist depending on the form of the nonlinearity and dispersion. They represent transient forms combining the properties of solitons, compactons, and peakons. It is shown that these solutions can exist against an inhomogeneous and nonstationary background.

Key words: KdV equation, mKdV equation, $\mathrm{K}(m,n)$ equation, Rosenau–Hyman equation, $\mathrm{K}(\cos)$ equation, Rosenau–Pikovsky equation, compacton, kovaton, soliton, peakon, peakocompacton.

DOI: https://doi.org/10.1134/S0044466919010149


English version:
Computational Mathematics and Mathematical Physics, 2019, 59:1, 150–159

Bibliographic databases:

UDC: 519.634
Received: 01.12.2017
Revised: 22.04.2018

Citation: S. P. Popov, “Compacton solutions of the Korteweg–de Vries equation with constrained nonlinear dispersion”, Zh. Vychisl. Mat. Mat. Fiz., 59:1 (2019), 158–168; Comput. Math. Math. Phys., 59:1 (2019), 150–159

Citation in format AMSBIB
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\paper Compacton solutions of the Korteweg--de Vries equation with constrained nonlinear dispersion
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2019
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\issue 1
\pages 158--168
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\pages 150--159
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