On Wave Propagation in Non-Linear Fields
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In this paper it is intended to investigate the propagation characters of waves in non-linear field with applications to the classical covariant field theories;the main interest is in discussions of singularities on wave fronts. Part I is concerned with the discussions of wave propagation in general non-linear field. As mathematical preliminaries the results of the characteristic theory of hyperbolic partial differential equations are summarized in §2 and §3. Readers who have elementary knowledges of the hyperbolic partial differential equations may omit §2;moreover, if they are familiar with the method of characteristics in hydrodynamics, §3 may be omitted. Since the discussions of these sections are based on physical considerations, the rigorous mathematical foundations of these discussions should be referred to standard text books of the partial differential equations. In §4, propagation of discontinuities along characteristic lines is investigated for general quasi-linear hyperbolic systems in two independent variables. As the result, a criterion for occurrence of discontinuity waves is given. In Part II, we deal with applications to the covariant field theories. Results obtained in §4 are applied directly to the investigation of occurrence of shock waves in gravitational field. In §6, solutions of non-linear scalar fields are analyzed in detail. In §7, the non-linear electrodynamics including the Born-Infeld theory is discussed. In §8, an example of derivative interactions between spinor fields is investigated in connection with the validity of micro-causality.
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