Numerical and Analytical Investigations of Nonlinear Synthesis Problems of Antenna with the Flat Radiating Aperture

The review article contains results of researches of the synthesis problems of radiating systems with incomplete input information arising in particular during optimal design of radio, acoustic or other types of radiating systems (RS) by the given requirements to energy characteristics of radiated field. In mathematical terms, these tasks are reduced to study and nume­rical solution of one class of two-dimensional nonlinear integral equations of Hammerstein type that depend on two real para­meters. It was established that a characteristic feature of this class of equations is nonuniqueness and the branching (or bifur­cation) of existing solutions. Methods of solving two-parametric nonlinear spectral problem, which is necessary to finding the set of points of branching are proposed. Algorithms and numerical methods for the finding of branching solutions are built and founded. Numerous examples of specific synthesis problems are presented.

Trapped modes around freely floating bodies in a two-layer fluid channel

Unlike the trapping of time-harmonic water waves by fixed obstacles, the oscillation of freely floating structures gives rise to a complex nonlinear spectral problem. Still, through a convenient elimination scheme the system simplifies to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we present conditions guaranteeing the existence of trapped modes in a two-layer fluid channel. Numerous examples of floating bodies supporting trapped modes are given.