Wave confinement, e.g., in waveguides, gives rise to a huge number of distinct phenomena. Among them, amplitude gain is a recurrent and relevant effect in undulatory processes. Using a general purpose protocol to solve wave equations, the boundary wall method, we demonstrate that for relatively simple geometries, namely, a few leaky or opaque obstacles inside a theta wedge waveguide (described by the Helmholtz equation), one can obtain a considerable wave amplification in certain spatially localized regions of the system. The approach relies on an expression for the wedge waveguide exact Green's function in the case of theta = pi /M (M = 1, 2, ...), derived through the method of images allied to group theory concepts. The formula is particularly amenable to numerical calculations, greatly facilitating simulations. As an interesting by-product of the present framework, we are able to obtain the eigenstates of certain closed shapes (billiards) placed within the waveguide, as demonstrated for triangular structures. Finally, we briefly discuss possible concrete realizations for our setups in the context of matter and electromagnetic (for some particular modes and conditions) waves.

The boundary wall method (BWM) is a general purpose protocol to treat boundary value problems for wave equations, specially Helmholtz’s (the case addressed here). Similarly to most approaches, the BWM may be computationally demanding for large borders C , at which the wave function must satisfy specified boundary conditions. Also, despite the fact the BWM is an exact procedure, usually it is not amenable to closed form solutions. The BWM relies on the Green’s function G 0 of the embedding domain V of C . However, in many instances—like for C modeling a billiard—the specific V is not really fundamental and thus one has a certain freedom to choose distinct domains and so G 0 ’s. Here we consider this characteristic of the BWM and show how to obtain some analytical results and solve numerically semi-infinite waveguides by exploring proper Green’s functions. As examples, we discuss rectangular, triangular and trapezoidal structures with both Dirichlet and leaking boundaries as well as scattering states within semi-infinite rectangular waveguides.

By continuously varying certain geometric parameters γ of the totally desymmetrized quantum Sinai billiard, we study the formation of the so-called soliton-like structures in the spectra of the resulting family of systems. We present a detailed characterization of the eigenstate ψn morphologies along such structures. Usually, scarring and bouncing ball mode states are expected to fully explain the solitons. However, we show that they do not exhaust all the possibilities. States with strong resemblance to very particular solutions of the associated integrable case ( 45°- 45° right triangle) also account for the ψn's. We argue that for the emergence of the solitons, in fact, there must be an interplay between the spatial localization properties of the soliton-related ψn's and the rescaling properties of the billiards with γ. This is illustrated, e.g., by comparing the behavior of the eigenwavelengths along the solitons and the billiard size dependence on γ. Considerations on how these findings could extend to other type of billiards are also briefly addressed.

The problem of Jeans gravitational instability is investigated for static and expanding universes within the context of the five and thirteen field theories which account for viscous and thermal effects. For the five-field theory a general dispersion relation has been derived with the help of relevant linearized perturbation equations, showing that the shear viscosity parameter alters the propagating modes for large and small wavelengths. The behavior of density and temperature contrasts are analyzed for the hard-sphere model in detail. In the small wavelengths regime, increasing the amount of shear viscosity into the system forces the harmonic perturbations to damp faster, however, in the opposite limit larger values of shear viscosity lead to smaller values of density and temperature contrasts. We also consider the hyperbolic case associated with the thirteen-field theory which involves two related parameters, namely the shear viscosity and the collision frequency, the last one is due to the production terms which appear in the Grad method. The dispersion relation becomes a polynomial in the frequency with two orders higher in relation to the five-field theory, indicating that the effects associated with the shear viscosity and heat flux are nontrivial. The profile of Jeans mass in terms of the temperature and number density is explored by contrasting with several data of molecular clouds. Regarding the dynamical evolution of the density, temperature, stress and heat flux contrasts for a universe dominated by pressureless matter, we obtain also damped harmonic waves for small wavelengths. In the case of large wavelengths, the density and temperature contrasts grow with time (due to the Jeans mechanism) while the stress and heat flux contrasts heavily decay with time. For an expanding universe, the Jeans mass and Jeans length are obtained and their physical consequences are explored.