Schmidt H.-J. Nonlinear wave equations

Universität Osnabrück, 2003. — 77 p.
This course addresses itself mainly to the fellows of the graduate college “Nonlinearities of optical materials”. It aims at providing a general physical background rather than dealing with the details of the various projects of this graduate college. It is generally held that most phenomena of nature are, in principle, nonlinear. Linearity is considered to occur only under idealized special circumstances. For example, any real oscillation will be nonlinear, and can only be approximated by a linear (or harmonic) oscillation in the case of small amplitudes. In contrast to this general situation, most methods of theoretical physics are confined to the linear case, which is mathematically much easier to handle than the nonlinear case. Yet the abundance of nonlinearity seems somewhat paradoxical in view of the fundamental theories of physics which are, at present, general relativity and quantum field theory, including gauge theories of the fundamental interactions. These are indeed nonlinear theories, but their limiting cases which are believed to describe condensed matter and electromagnetic waves, are linear. Among the limiting cases we have the quantum theory of atoms, molecules and solid states as well as electrodynamics, quantum or classical.