Stability of nonstationary solutions of the generalized KdV-Burgers equation.

*(English. Russian original)*Zbl 1321.35196
Comput. Math. Math. Phys. 55, No. 2, 251-263 (2015); translation from Zh. Vychisl. Mat. Mat. Fiz. 55, No. 2, 253-266 (2015).

Summary: The stability of nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters.

##### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74J10 | Bulk waves in solid mechanics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

generalized KdV-Burgers equation; stability of nonstationary solutions; difference solution method
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\textit{A. P. Chugainova} and \textit{V. A. Shargatov}, Comput. Math. Math. Phys. 55, No. 2, 251--263 (2015; Zbl 1321.35196); translation from Zh. Vychisl. Mat. Mat. Fiz. 55, No. 2, 253--266 (2015)

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