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Introduction to nonclassical shock solutions of the modified Burgers‐Korteweg‐de Vries equation
Author(s) -
Kluwick A.
Publication year - 2002
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/1617-7061(200203)1:1<55::aid-pamm55>3.0.co;2-k
Subject(s) - nonlinear system , burgers' equation , shock wave , hyperbolic partial differential equation , dissipation , classification of discontinuities , mathematical analysis , physics , shock (circulatory) , scalar (mathematics) , convexity , scalar field , classical mechanics , mathematics , mechanics , geometry , quantum mechanics , medicine , financial economics , economics
A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg‐de Vries‐Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two‐layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.