Acoustic Scattering and the Extended Korteweg– de Vries Hierarchy

The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa–Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

Ultrasonic measurement of the diffusion bond strength

This paper presents an ultrasonic method for measurement of diffusion bond strength between two identical materials. The method requires a single normal-incidence ultrasonic measurement. When a diffusion bond is not perfect some ultrasonic energy is reflected from the interface separating the two substrates. Amplitude of a reflected signal correlates qualitatively with the bond strength. However, a quantitative correlation is difficult to establish since reflected signal amplitude depends on parameters of the test equipment. We introduce a spectroscopic analysis of a reflected signal to measure the bond strength. We show experimentally that an imperfect diffusion bond can be characterized by a single parameter: interfacial spring stiffness. The stiffness can be determined from the reflected signal spectrum and does not depend on testing equipment. It is also demonstrated that for a particular diffusion bonding procedure as applied to tool steel the interfacial stiffness correlates quantitatively with bond strength.     

Alpha spectrometric evaluation of SRM-995 as a potential uranium/thorium double tracer system for age-dating uranium materials

Uranium-233 (t _1/2 ~ 1.59E5 years) is an artificial, fissile isotope of uranium that has significant importance in nuclear forensics. The isotope provides a unique signature in determining the origin and provenance of uranium-bearing materials and is valuable as a mass spectrometric tracer. Alpha spectrometry was employed in the critical evaluation of a ²³³U standard reference material (SRM-995) as a dual tracer system based on the in-growth of ²²⁹Th (t _1/2 ~ 7.34E3 years) for ~35 years following radiochemical purification. Preliminary investigations focused on the isotopic analysis of standards and unmodified fractions of SRM-995; all samples were separated and purified using a multi-column anion-exchange scheme. The ²²⁹Th/²³³U atom ratio for SRM-995 was found to be 1.598E?4 (±4.50 %) using recovery-corrected radiochemical methods. Using the Bateman equations and relevant half-lives, this ratio reflects a material that was purified ~36.8 years prior to this analysis. The calculated age is discussed in contrast with both the date of certification and the recorded date of last purification.     

Calogero–FranÇoise Flows and Periodic Peakons

The completely integrable Hamiltonian systems discovered by Calogero and FranÇoise contain the finite-dimensional reductions of the Camassa–Holm and Hunter–Saxton equations. We show that the associated spectral problem has the same form as that of the periodic discrete Camassa–Holm equation. The flow is linearized by the Abel map on a hyperelliptic curve. For two-particle systems, which correspond to genus-1 curves, explicit solutions are obtained in terms of the Weierstrass elliptic functions.     

Low regularity local solutions for field equations

We prove local existence and uniqueness of low regularity solutions to semi-linear systems. The Smoothness of our solutions is below the classical level. The main lemma is an L² estimate for products of solutions of linear equations inspried by earlier work by Strichartz, Klainerman and Machedon.