Abstract:
Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focussed on long-range propagation in deep ocean environments. For this class of problems, the ray equations may be simplified by making use of a one-way formulation in which the range variable appears as the independent (time-like) variable. Topics discussed include integrable and non-integrable ray systems, action-angle variables, nonlinear resonances and the KAM theorem, ray chaos, Lyapunov exponents, predictability, nondegeneracy violation, ray intensity statistics, semiclassical breakdown, wave chaos, and the connection between ray chaos and mode coupling. The Hamiltonian structure of the ray equations plays an important role in all of these topics.

Abstract:
The inverse problem of finding the refraction coefficient of a shallow ocean from the observation of the extra Cauchy data for the acoustic field at the surface of the ocean is studied. Uniqueness theorem is proved and a reconstruction algorithm is obtained. Comments are given on the related results in the literature.

Abstract:
We discuss how the Perfectly Matched Layer (PML) can be adapted to numerical simulations of nonlinear and matter wave systems, such as Bose-Einstein condensates. We also present some examples which illustrate the benefits of using the PML in the simulation of nonlinear and matter waves.

Abstract:
The perfectly matched layers (PMLs), as a boundary termination over an unbounded spatial domain, are widely used in numerical simulations of wave propagation problems. Given a set of discretization parameters, a procedure to select the PML profiles based on minimizing the discrete reflectivity is established for frequency domain simulations. We, by extending the function class and adopting a direct search method, improve the former procedure for traveling waves.

Abstract:
We derive periodic multilayer absorbers with effective uniaxial properties similar to perfectly matched layers (PML). This approximate representation of PML is based on the effective medium theory and we call it an effective medium PML (EM-PML). We compare the spatial reflection spectrum of the layered absorbers to that of a PML material and demonstrate that after neglecting gain and magnetic properties, the absorber remains functional. This opens a route to create electromagnetic absorbers for real and not only numerical applications and as an example we introduce a layered absorber for the wavelength of $8$~$\mu$m made of SiO$_2$ and NaCl. We also show that similar cylindrical core-shell nanostructures derived from flat multilayers also exhibit very good absorptive and reflective properties despite the different geometry.

Abstract:
We construct a perfectly matched absorbing layer for stationary Schrodinger equation with analytic slowly decaying potential in a periodic structure. We prove the unique solvability of the problem with perfectly matched layer of finite length and show that solution to this problem approximates a solution to the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity.

Abstract:
Deep ocean acoustics, in the absence of shipping and wildlife, is driven by surface processes. Best understood is the signal generated by non-linear surface wave interactions, the Longuet-Higgins mechanism, which dominates from 0.1 to 10 Hz, and may be significant for another octave. For this source, the spectral matrix of pressure and vector velocity is derived for points near the bottom of a deep ocean resting on an elastic half-space. In the absence of a bottom, the ratios of matrix elements are universal constants. Bottom effects vitiate the usual "standing wave approximation," but a weaker form of the approximation is shown to hold, and this is used for numerical calculations. In the weak standing wave approximation, the ratios of matrix elements are independent of the surface wave spectrum, but depend on frequency and the propagation environment. Data from the Hawaii-2 Observatory are in excellent accord with the theory for frequencies between 0.1 and 1 Hz, less so at higher frequencies. Insensitivity of the spectral ratios to wind, and presumably waves, is indeed observed in the data.

Abstract:
We propose an Adaptive Perfectly Matched Layer (APML) to be used in diffraction grating modeling. With a properly tailored co-ordinate stretching depending both on the incident field and on grating parameters, the APML may efficiently absorb diffracted orders near grazing angles (the so-called Wood's anomalies). The new design is implemented in a finite element method (FEM) scheme and applied on a numerical example of a dielectric slit grating. Its performances are compared with classical PML with constant stretching coefficient.

Abstract:
The perfectly matched layers (PML) and exterior complex scaling (ECS) methods for absorbing boundary conditions are analyzed using spectral decomposition. Both methods are derived through analytical continuations from unitary to contractive transformations. We find that the methods are mathematically and numerically distinct: ECS is complex stretching that rotates the operator's spectrum into the complex plane, whereas PML is a complex gauge transform which shifts the spectrum. Consequently, the schemes differ in their time-stability. Numerical examples are given.