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Nonlinear algebraic analysis of delta shock wave solutions to Burgers’ equation. (English) Zbl 1059.35122
This paper uses an abstract algebraic analysis to study delta shock wave solutions to Burgers’ equation. It uses three fundamental structures in order to define in a very general way a sheaf of differential algebras that contains within it most of the special cases that are encountered in the theory of generalized functions.
By making a suitable choice of structures, Burgers’ equation with a Dirac \(\delta\) measure as initial data is considered, and a generalized \(\delta\)-shock wave is constructed that forms an approximate solution that is self-similar to the initial data.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35L60 First-order nonlinear hyperbolic equations
58J47 Propagation of singularities; initial value problems on manifolds
76L05 Shock waves and blast waves in fluid mechanics
46F99 Distributions, generalized functions, distribution spaces
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