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Licentiate seminar

Theoretical studies of shock waves in dispersive and dissipative media


Defendant Main Advisor Extra Advisor Date
Henrik Sandqvist Bengt Enflo 2001-05-07

Opponent
Oleg Rudenko, Moscow

Evaluation committee

Abstract

Propagation of waves in nonlinear, dispersive and dissipative media, as described by Korteweg-de Vries-Burgers` equation (KdVB), has been studied. The focus of the investigation has been to study analytically, the structure of a shock wave that is broken down by dispersive and dissipative phenomena. To be able to use the inverse scattering transform (IST) to get analytical solutions for Korteweg-de Vries` equation (KdV), an N-wave was used as model for the initial shock. The IST is used to transform KdV, which is a nonlinear differential equation, into Marchenko`s equation that is a linear Volterra integral equation. A zeroth order iteration solution, which reconstructs the initial waveform, is presented. For positive times, this solution shows a decaying shock front which slows down, leaving an oscillating tail behind. This solution is valid for moderate values of the dispersion coefficient. In order to obtain solutions for smaller values of the dispersion coefficient, asymptotic analysis is used. The corresponding asymptotic analysis for Burgers` equation, with a small dissipation coefficient, is quoted for comparison. An asymptotic analysis is also made for KdVB, in the case for which dispersion as well as dissipation is important.
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