Mathematical Sciences: Nonlinear Problems in Classical Electromagnetic Theory

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8823037 Bloom This proposal is for a continuation of work on models of wave propagation in nonlinear dielectric media and on problems associated with the behavior of nonlinear transmission lines. The mathematical formulations involve partial differential equations in one and several dimensions of both hyperbolic and mixed type; the interest is in establishing results on the local and global existence of solutions to Cauchy and initial-boundary value problems, as well as results on the asymptotic behavior of solutions. Also considered are the related problems of breakdown of smooth solutions, shock formation, and the existence of shock surfaces. Many of the models presented are new (e.g., wave propagation in a nonlinear superconducting media and the multidimensional wave propagation problems in dielectrics with nontrivial conductivity) and several lead to problems in which a physical parameter tends to zero.