Nonlinear Waves in Networks

This volume studies existence, regularity, and uniqueness of local solutions in time for quasi-linear wave equations on one-dimensional networks with linear transmission conditions. A unified description on interactions between media with time evolution, proposed by the author is outlined.

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The main topic of this text is the well-posedness locally in time, of quasilinear wave equations on one-dimensional networks with linear transmission conditions. Such problems arise for example in the modelling of transversal vibrations of networks of strings. The setting serves as a model case for the systematic study, with T. Kato's theory, of the phenomenon of nonlinear compatibility conditions arising if regular solutions are desired. This phenomenon is bound to occur always if hyperbolic evolution is combined with all kinds of coupling conditions. The connections of the results to a concept, previously proposed by the author, for the description of complicated interactions between media (of possibly varying space dimension) with time evolution, are described. A strategy for physical legitimations of linear mechanical coupling models, the systematic derivation of physical laws and of the asymptotics of frequencies of eigenmodes are outlined. As applications of this interaction concept, small vibrations of membranes with conic boundary-points and a membrane-string-coupling are considered.