Propagation and interaction of singularities in nonlinear hyperbolic problems / Michael Beals

The book gives an overview of the use of microlocal analysis and commutators in the study of solutions to nonlinear wave equations. In the analysis of solutions to linear problems, pseudo-differential operators are an important tool since they allow the decomposition of the singular part of solutions into pieces which can be analyzed separately and then reassembled. When a nonlinear function acts on a solution, the interaction of the microlocal pieces complicates the procedure, but in many cases the analysis can be successfully carried out. Beginning with previous works of Rauch and Bony, the tools of microlocal analysis have been applied to the study of nonlinear problems in an attempt to describe the propagation of singularities and of regularity for solutions. Certain results analogous to those of the linear theory continue to hold, but purely nonlinear phenomena are also known to occur. In the book a number of the developments in this nonlinear theory of microlocal singularities are presented (zbMath)