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On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

Trapeznikov Institute of Control Scienties, Russian Academy of Sciences, 65 Profsoyuznaya Street, 117997 Moscow, Russia
Academic Editors: Jan Awrejcewicz and Valentin Lychagin
Symmetry 2021, 13(2), 220; https://doi.org/10.3390/sym13020220
Received: 22 December 2020 / Revised: 23 January 2021 / Accepted: 26 January 2021 / Published: 29 January 2021
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations)
We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found. View Full-Text
Keywords: Korteweg–de Vries–Burgers equation; cylindrical and spherical waves; saw-tooth solutions; periodic boundary conditions; head shock wave Korteweg–de Vries–Burgers equation; cylindrical and spherical waves; saw-tooth solutions; periodic boundary conditions; head shock wave
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MDPI and ACS Style

Samokhin, A. On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations. Symmetry 2021, 13, 220. https://doi.org/10.3390/sym13020220

AMA Style

Samokhin A. On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations. Symmetry. 2021; 13(2):220. https://doi.org/10.3390/sym13020220

Chicago/Turabian Style

Samokhin, Alexey. 2021. "On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations" Symmetry 13, no. 2: 220. https://doi.org/10.3390/sym13020220

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