Light scattering by a small spherical particle with a low dissipation rate is discussed based upon the Mie theory. It is shown that if close to the plasmon (polariton) resonance frequencies the radiative damping prevails over dissipative losses, sharp giant resonances with very unusual properties may be observed. In particular, the resonance extinction cross section increases with an increase in the order of the resonance (dipole, quadrupole, etc.); the characteristic values of electric and magnetic near fields for the scattered light are singular in the particle size, while energy circulation in the near field is rather complicated, so that the Poynting vector field includes singular points whose number, types, and positions are very sensitive to fine changes in the incident light frequency. The results may provide new opportunities for a giant, controlled, highly frequency-sensitive enhancement and variation of electromagnetic field at nanoscales.

Domain boundaries (DB’s) in convection patterns are studied near the onset of convection within the framework of amplitude equations of the Newell-Whitehead-Segel type. It is demonstrated that DB’s of an arbitrary direction are possible between domains occupied by convection rolls with different orientations. Those DB’s are always immobile, and they perform perfect selection of the rolls’ wave numbers. In certain particular cases, exact solutions for DB’s of this type are found, and their stability against long-wave flexural perturbations is established. DB’s between the rolls and a hexagonal pattern in the slightly overcritical case, and between the hexagons and the conduction (quiescent) state in the slightly subcritical case, are investigated as well. In this case, the DB is, generally speaking, moving with a constant velocity. The exponentially weak effect of pinning of the moving DB by a small-scale underlying structure is studied. Formation of DB’s of the ‘‘quiescent-state–hexagon’’ and ‘‘roll-hexagon’’ types near a sidewall is also investigated. The special case of a DB between two systems of rolls, the angle between which is close to π/3 or 2π/3, is studied in detail. This DB is, as a matter of fact, a bound state of two DB’s of the ‘‘roll-hexagon’’ type. Finally, it is demonstrated that in one-dimensional patterns (rolls) a stationary DB is not possible. In this case, a dynamical problem of decay of a DB-like structure is solved.

The conditions for observing Fano resonances at elastic light scattering by a single finite-size obstacle are discussed. General arguments are illustrated by consideration of the scattering by a small (relative to the incident light wavelength) spherical obstacle based upon the exact Mie solution of the diffraction problem. The most attention is paid to recently discovered anomalous scattering. An exactly solvable one-dimentional discrete model with nonlocal coupling for simulating diffraction in wave scattering in systems with reduced spatial dimensionality is also introduced and analyzed. Deep connections between the resonances in the continuous and discrete systems are revealed.

We present the results of a detailed analytical study of light scattering by a particle with high refractive index $m+i\kappa$ and low losses ($m\gg 1,0<\kappa \ll 1$) based on the exact Mie solution. We show that there is a dramatic difference in the behavior of the electromagnetic field within the particle (inner problem) and outside it (outer problem). With an increase in $m$ at fixed values of the other parameters, the field within the particle asymptotically converges to a periodic function of $m$. The electric and magnetic type Mie resonances of different orders overlap substantially. It may lead to a giant concentration of the electromagnetic energy within the particle. At the same time, we demonstrate that the solution for the outer problem makes it possible to present each partial scattered wave as a sum of two partitions. One of them corresponds to the $m$-independent wave, scattered by a perfectly reflecting particle and plays the role of a background, while the other is associated with the excitation of a sharply $m$-dependent resonant Mie mode. The interference of the partitions brings about a typical asymmetric Fano profile. The profile is obtained from the exact Mie solution by means of identical transformations without any additional assumptions and/or fitting. It makes it possible to generalize rigorously the Fano theory to the case of finite dissipation. At an increase in $m$ the Fano resonances in the outer problem die out and the scattered field converges to the universal, $m$-independent profile. The behavior of the resonances at a fixed $m$ and varying particle size parameter $\left(x\right)$ is also discussed in detail. The similarities and differences of the two cases (fixed $x$, varying $m$ and fixed $m$, varying $x$) are disclosed. We also show that under certain very general conditions the scattering cross section of a large lossy sphere cannot be smaller than half its geometric cross section, while its absorption cross section cannot exceed three halves of the geometric one. Numerical estimates of most discussed effects for a gallium phosphide particle irradiated by the second harmonic of a Nd:YAG laser are presented as an example. In addition to purely academic interest, the obtained results may be employed to design new highly nonlinear heterogenic nanostructures and other metamaterials.

The future of ultra-fast optical communication systems is inevitably connected with progress in optical circuits and nanoantennas. One of the key points of this progress is the creation of elementary components of optical devices with scattering diagrams tailored for redirecting the incident light in a desired manner. Here we demonstrate theoretically and experimentally that a small, simple, spatially homogeneous dielectric subwavelength sphere with a high refractive index and low losses (as some semiconductors in the visible or near infrared region) exhibits properties allowing to utilize it as a new multifunctional element for the mentioned devices. This can be achieved by taking advantage of the coherent effects between dipolar and multipolar modes, which produce anomalous scattering effects. The effects open a new way to control the directionality of the scattered light. The directional tuning can be obtained in a practical way just by a change in the frequency of the incident wave, and/or by a well-chosen diameter of the sphere. Dielectric nanoparticles with the required optical properties in the VIS-NIR may be now readily fabricated. These particles could be an efficient alternative to the widely discussed scattering units with a more complicated design.

Numerical study of the one-dimensional nonlinear partial differential equation, equivalent to that proposed [Recent Advances in Engineering Science (Springer-Verlag, Berlin, 1989)] to describe longitudinal seismic waves, is presented. The equation has a threshold of short-wave instability and symmetry, providing slow long-wave dynamics. It is shown that the threshold of the short-wave instability corresponds to a point of “continuous” (second order) transition from a spatially uniform state to a chaotic regime. Thus, contrary to the conventional scenarios, turbulence arises from the spatially uniform state as a result of one and the only one supercritical bifurcation.