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Characteristics of differential systems and wave propagation. Reprint of the 1931 orig. (Caratteristiche dei sistemi differenziali e propagazione ondosa. Lezioni raccolte da G. Lampariello.) (Italian) Zbl 0784.35002

Bologna: Zanichelli Editore. vii, 111 p. (1988).
Reprint of the original edition (1931), for a review see JFM 57.0541.01.
These lectures are concerned with the problem and the method of Cauchy related to systems of partial differential equations of first order in \(n\) independent variables. The main object is the analysis of the wave propagation by means of classical results of Hadamard, Hamilton, Jacobi and Hugoniot. Various examples from hydrodynamic and electromagnetic theories are illustrated in details. The treatment of the subject is described with the clearness and the rigour typical for this Italian scientist.
At first, the existence theorem of Cauchy-Kowalewsky is outlined and characteristic manifolds, together with discontinuities of solutions are discussed. As a first example, the three dimensional wave equation is examined. Successively, the author expounds Cauchy’s method for general first order partial differential equations in \(n\) independent variables and the construction of envelopes and characteristic curves. Examples from fluid mechanics are specified. Further, various aspects of the wave propagation related to the Maxwell equations are deduced (Fresnel’s wave surfaces). At last, a short account of Hamilton’s function of the bicharacteristics for the Schrödinger equation is given, too.
Reviewer: P.Renno (Napoli)

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
01A75 Collected or selected works; reprintings or translations of classics
35A10 Cauchy-Kovalevskaya theorems
35F20 Nonlinear first-order PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory

Citations:

JFM 57.0541.01
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