This article is cited in 9 scientific papers (total in 9 papers)
Stability of discontinuity structures described by a generalized KdV–Burgers equation
A. P. Chugainovaa, V. A. Shargatovb
a Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia
b National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, Moscow, 115409, Russia
The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form $\varphi(u)=u^4-u^2$ is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.
generalized KdV–Burgers equation, spectral (linear) stability of stationary solutions, special discontinuities.
|Russian Science Foundation
|Chugainova acknowledges the support of the Russian Science Foundation, project no. 14-50-00005.
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Computational Mathematics and Mathematical Physics, 2016, 56:2, 263–277
A. P. Chugainova, V. A. Shargatov, “Stability of discontinuity structures described by a generalized KdV–Burgers equation”, Zh. Vychisl. Mat. Mat. Fiz., 56:2 (2016), 259–274; Comput. Math. Math. Phys., 56:2 (2016), 263–277
Citation in format AMSBIB
\by A.~P.~Chugainova, V.~A.~Shargatov
\paper Stability of discontinuity structures described by a generalized KdV--Burgers equation
\jour Zh. Vychisl. Mat. Mat. Fiz.
\jour Comput. Math. Math. Phys.
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A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113
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