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A multi-scale approach to hyperbolic evolution equations with limited smoothness. (English) Zbl 1173.35130

The authors show how one can study the solutions of a certain class of hyperbolic evolution equations with limited smoothness, using techniques from so-called multiresolution analysis. These equations are of the type
\[ \left[ \partial_{z}-iP(z,x,D_{x})\right] u=0, \]
where \(z\) is an evolution parameter (e.g. time). When \(P\) is a pseudodifferential operator with smooth real symbol of order one, then it is well-known that the solution is given by a Fourier integral operator. The authors pose the question “how can we express the solutions” when it is assumed that the symbol of \(P\) has limited smoothness? The specific method they use to attack the problem has its roots in the theory of coherent wavepackets and Fourier integral operators, and uses paradifferential decompositions of functions. The scattering between wavepackets being described by a Volterra equation of the second kind. Applications to wave-equation imaging problems are given, and an interesting example involving seismic data downward continuation is considered. In addition the authors describe a new computational algorithm.

MSC:

35S30 Fourier integral operators applied to PDEs
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
86A15 Seismology (including tsunami modeling), earthquakes
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