On the global existence and wave-breaking criteria for the two-component Camassa-Holm system

We discuss the foundations and state-of-the-art of the method of analytical regularization (MAR) (also called the semi-inversion method). This is a collective name for a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second-kind integral equation with a smoother kernel, to ensure point-wise convergence of the usual discretization schemes. This is done using analytical inversion of a singular part of the original equation; discretization and semi-inversion can be combined in one operation. Numerous problems being solved today with this approach are reviewed, although in some of them, MAR comes in disguise.

The method of analytical regularization in wave-scattering and eigenvalue problems: foundations and review of solutions

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

High-frequency asymptotic solutions of electromagnetic edge diffraction by two different theories are studied. One is the uniform asymptotic theory which is a refinement of Keller's geometrical theory of diffraction. The other is Ufimtsev's theory of the edge wave, representing an improvedment over the classical physical optics theory. These two theories are summarized, their features compared, and their relations discussed. When the observation point is away from shadow boundaries and caustics, both uniform asymptotic theory and Ufimtsev's theory, up to (and including) order k^{-1/2} , agree with Keller's theory. Near or on shadow boundaries, uniform asymptotic theory gives an explicit field solution, while Ufimtsev's result contains a physical optics integral. The evaluation of that integral is not a trivial task. In a two-dimensional test problem, it is shown that both theories do give, up to order k^{-1/2} , an identical field solution everywhere including the edge, shadow boundaries, and transition regions.

Comparison of uniform asymptotic theory and Ufimtsev's theory of electromagnetic edge diffraction

The knowledge of high-frequency diffraction of an arbitrary wave incident on an edge is important in many applications, such as antennas mounted on aircraft and reflector antennas illuminated by complex feeds. In this paper the problem of a half plane illuminated by a nonplanar wave is investigated using the concept of the plane wave spectral representation. For large wave number k, a new higher-order asymptotic solution for the total field up to and including terms of order k−5/2 relative to the incident field is derived. The behavior of the solution for the observation points which coincide with shadow boundary directions of a multipole line source is discussed in detail. Furthermore, numerical solution of the field integral representation is constructed for the observation angles in the transition regions. The results are compared with those of the Geometrical Theory of Diffraction (GTD), the Uniform Asymptotic Theory (UAT), the Uniform Theory of Diffraction (UTD) and the Modified Slope Diffraction (MSD).

Spectral analysis of high‐frequency diffraction of an arbitrary incident field by a half plane—Comparison with four asymptotic techniques

Our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. We assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains. Fairly standard methods give existence of a minimum (not necessarily unique) with e.g., L2-regularity. The assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal “necessary conditions for optimality.” A new general technique is developed to justify these conditions; it then follows that minimizers satisfy the equilibrium conditions in the classical sense. (A feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class.) In the symmetric case, which admits trivial (straight and untwisted) solutions, we show that nontrivial solutions also exist if the field is strong enough.

Equilibrium states of an elastic conducting rod in a magnetic field

Let θ be an open set in the plane which contains the interval I: {(x, y)∣y=0, -1?x?1}. We consider functions u, which are harmonic in θ-I and continuous in θ. Then, without additional smoothness conditions on f(x)=u(x,0),-1?x$le;1, the one sided normal derivatives(?u/?y)+limh?0+u(x,h)-f(x)/h and (?u/?y)-=limh?0-u(x,h)-f(x)/h, may not exist at any point of I. Here we assume only that f(x) is continuous. We show that in this case the normal derivatives will exist in a "Sobolev-like" Space of distributions.

Some results on the boundary regularity of harmonic functions

Bifurcation theory of an elastic conducting rod in a magnetic field

The effect of bending stiffness on inextensible cables

Diffraction of plane waves by a semicircular strip

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