Mean value properties of solutions to the m-dimensional Helmholtz and modified equations are considered. An elementary derivation these is given. It involves Euler–Poisson–Darboux equation. Despite similar form for both equations, their consequences distinguish essentially. The restricted mean property harmonic functions amended so that a function satisfying this in bounded domain special class...

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at certain wavenumbers (those that correspon...

We perform numerical simulations of the Kelvin-Helmholtz instability in the mid-plane of a protoplanetary disk. A two-dimensional corotating slice in the azimuthal–vertical plane of the disk is considered where we include the Coriolis force and the radial advection of the Keplerian rotation flow. Dust grains, treated as individual particles, move under the influence of friction with the gas, wh...

New families of exact analytical Helmholtz solitons are presented for various material nonlinearities, and simulations confirm their stability. A novel generalization of Snell’s Law for Kerr solitons is also reported. Helmholtz angular corrections to paraxial theory, exceeding 100%, are uncovered. © 2007 Optical Society of America OCIS codes: (190.4400) Nonlinear Optics, Materials; (190.9540) N...