A parabolic equation for penetrable rough surfaces: using the Foldy-Wouthuysen transformation to buffer density jumps

The goal of this effort is to design a parabolic equation (PE) that can be fully integrated into modern rough surface scattering theory. The Foldy-Wouthuysen transformation is used to design a PE that addresses this challenge. The paradigm employed is based on a non-relativistic theory of the quantum Lamb shift, in which a parabolic equation (the Schrödinger equation) was used to model a field near a rough surface (the world line of the hydrogen nucleus advected by vacuum fluctuations). With the acoustic field serving as the prototypical classical field, the PE derived using the Foldy-Wouthuysen transformation exploits higher-order boundary conditions to buffer density discontinuities in a manner precisely dictated by the formalism. This is significant because the techniques used in rough surface scattering theory ultimately rely on perturbation theory, conformal mappings, a local method of images, or some similar distortion of the range-independent problem. This type of distortion is not possible with techniques currently employed at a density jump, and so ad hoc rules have been used instead. The new PE allows interfaces where the density jumps (such as the ocean bottom) to be distorted into rough ones, and so it is fully compatible with rough surface scattering theory.     

An analysis for imperfect sensitivity of a superconducting rod

On the basis of the interaction between the electromagnetic field and a solid, the imperfect sensitivity of a superconducting rod is studied in this paper. The influences of the initial deflection of the rod and the initial magnetic field on the stability are discussed. One can see that the bifurcation response of the rod with imperfections is similar to the universal unfolding of the fork bifurcation from the singularity theory.Project supported by the Foundation of the State Education Commission of China and the Natural Science Foundation of Gansu Province.