By Yu.A. Kravtsov
Caustics, Catastrophes and Wave Fields in a feeling keeps the remedy of the sooner quantity 6 "Geometrical Optics of Inhomogeneous Media" through analysing caustics and their fields at the foundation of contemporary disaster idea. the current quantity covers neighborhood and uniform caustic asymptotic expansions: The Lewis-Kravtsov approach to commonplace features, Maslov's approach to canonical operators , Orlov's approach to interference integrals, in addition to their transformations for penumbra, space-time, random and different kinds of caustics. all of the tools are amply illustrated by means of labored difficulties touching on suitable wave-field purposes.
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Additional info for Caustics, Catastrophes and Wave Fields
In the neighborhood of each of the two singular points, a smooth transformation reduces the function f(x, y) = 0 locally to a second-order polynomial (bottom diagram). The linear (fl = YI - xd and square (f2 = XI + y:) polynomials render examples of the normal forms to which virtually all curves in the x, y plane may be reduced. Exclusions are some degenerate cases corresponding to structurally unstable maps. An example of a structurally unstable mapping is displayed in the top diagram of Fig. 3d.
If the dependence of Y on x is bivalued, as shown in the top diagram of Fig. I 3J 3J 0) c) d) Fig. 3. Simple typical situations in mapping a smooth curvef(x, y) = 0 onto the horizontal axis. (a) One-to-one map. (b) Map with one critical point. (e) Map with two critical points. (d) Structurally unstable (nontypical) situation (solid lines) which is reduced by small perturbations (dashed lines) to either (a) or (c) f2(XI, yd = XI + yi = 0 (bottom diagram). , the curve f2(Xb yd = 0 has a vertical tangent, much like the original curve f(x, y) = o.
Close objects differ from the original by small perturbations or motions. , then small motions imply that both the functions and their derivatives are close. 2]. Transitions from one normal form to another via a structurally unstable state correspond to bifurcations in dynamic system theory. For example, a crossover from one-ray mode to three-ray mode observed in increasing y in Fig. 2b occurs at the point of bifurcation Yo. While all singularities that occur in mappingj(x, y) = 0 onto the x axis can be classed by common analysis, in a multidimensional case, classification is markedly complicated by the fact that the typical polynomials are also multidimensional.
Caustics, Catastrophes and Wave Fields by Yu.A. Kravtsov