By F. G. Friedlander
This booklet was once initially released in 1975. In Einstein's normal conception of Relativity the consequences of gravitation are represented via the curvature of space-time. actual procedures taking place within the presence of gravitation needs to then be handled mathematically by way of their behaviour in a curved space-time. the most simple of those strategies is wave propagation, and this ebook offers a rigourous dialogue of the neighborhood results of curvature at the behaviour of waves. during this dialogue many recommendations are built that are additionally wanted for a examine of extra basic difficulties, during which the gravitational box itself performs a dynamical position. even though a lot of the ebook bargains with 4-dimensional space-time, the n-dimensional case is additionally handled, extra in brief. The subject-matter is usually of curiosity in different branches of mathematical physics and, as a clean account of the classical paintings of Hadamard and M. Riesz, within the conception of partial differential equations
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Additional resources for The wave equation on a curved space-time
2. SOLITONS IN PERIODIC HETEROGENEOUS MEDIA 19 are exceptional models, in the sense that any additional term, which takes into regard physical effects that were not included in the basic model, breaks the exact integrability. This circumstance suggest the necessity to investigate solitons in nonintegrable models (in a strict mathematical sense, these solutions are not "solitons", but rather "solitary waves"; however, following commonly adopted practice, they will be called solitons). In many physically relevant situations, the additional terms which break the integrability of the unperturbed model are small, making it natural to apply a perturbation theory relying on an asymptotic expansion around exact solutions provided by the 1ST in the absence of the perturbations.
L 1 L. 10: A typical example of the shape (shown on the logarithmic scale) of dispersion-managed solitons in the ideal lossless model, and in its realistic counterpart with lumped filters and amplifiers (for comparison, the solitons with equal amplitudes are taken in both cases). Each soliton is shown at a position (close to the midpoint of the anomalous-GVD segment of the DM map) where it is narrowest. 45) (the latter is usually referred to as the Gaussian transfer function with the bandpass width Aw).
It is composed of alternating BGs with opposite signs of the Kerr nonlinearity, and also gives rise to a family of robust solitons. A periodic heterogeneous system with the x^^^ nonlinearity was proposed in the form of the so-called tandem model , which combines linear segments and ones carrying the quadratic nonlinearity . Specific solitons were revealed by numerical simulations in this model. In all these systems, stable solitons exist in the form of periodically oscillating breathers (obviously, the solitons cannot keep a permanent shape propagating through the inhomogeneous structure).