282 research outputs found
Localized Faraday patterns under heterogeneous parametric excitation
Faraday waves are a classic example of a system in which an extended pattern
emerges under spatially uniform forcing. Motivated by systems in which uniform
excitation is not plausible, we study both experimentally and theoretically the
effect of heterogeneous forcing on Faraday waves. Our experiments show that
vibrations restricted to finite regions lead to the formation of localized
subharmonic wave patterns and change the onset of the instability. The
prototype model used for the theoretical calculations is the parametrically
driven and damped nonlinear Schr\"odinger equation, which is known to describe
well Faraday-instability regimes. For an energy injection with a Gaussian
spatial profile, we show that the evolution of the envelope of the wave pattern
can be reduced to a Weber-equation eigenvalue problem. Our theoretical results
provide very good predictions of our experimental observations provided that
the decay length scale of the Gaussian profile is much larger than the pattern
wavelength.Comment: 10 pages, 9 figures, Accepte
Multiplicity and concentration results for local and fractional NLS equations with critical growth
Goal of this paper is to study the following singularly perturbed nonlinear Schrödinger equation: eps^2s (-Delta)^s v + V(x)v = f(v), x in R^N, where s is in (0,1), N is greater or equal to 2, V in C(R^N,R) is a positive potential and f is assumed critical and satisfying general Berestycki-Lions type conditions. When eps is greater than 0 is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of V; in particular, we improve the result in [37]. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting s = 1 and N greater or equal to 3, with an exponential decay of the solutions
Mixed diffusive-convective relaxation of a broad beam of energetic particles in cold plasma
We revisit the applications of quasi-linear theory as a paradigmatic model
for weak plasma turbulence and the associated bump-on-tail problem. The work,
presented here, is built around the idea that large-amplitude or strongly
shaped beams do not relax through diffusion only and that there exists an
intermediate time scale where the relaxations are convective (ballistic-like).
We cast this novel idea in the rigorous form of a self-consistent nonlinear
dynamical model, which generalizes the classic equations of the quasi-linear
theory to "broad" beams with internal structure. We also present numerical
simulation results of the relaxation of a broad beam of energetic particles in
cold plasma. These generally demonstrate the mixed diffusive-convective
features of supra-thermal particle transport; and essentially depend on
nonlinear wave-particle interactions and phase-space structures. Taking into
account modes of the stable linear spectrum is crucial for the self-consistent
evolution of the distribution function and the fluctuation intensity spectrum.Comment: 25 pages, 15 figure
Soliton generation and control in engineered materials
Optical solitons provide unique opportunities for the control of light‐bylight. Today, the field of soliton formation in natural materials is mature, as the
main properties of the possible soliton states are well understood. In particular, optical solitons have been observed experimentally in a variety of materials and physical settings, including media with cubic, quadratic, photorefractive, saturable, nonlocal and thermal nonlinearities.
New opportunities for soliton generation, stability and control may become accessible in complex engineered, artificial materials, whose properties
can be modified at will by, e.g., modulations of the material parameters or the application gain and absorption landscapes. In this way one may construct
different types of linear and nonlinear optical lattices by transverse shallow modulations of the linear refractive index and the nonlinearity coefficient or
complex amplifying structures in dissipative nonlinear media. The exploration of the existence, stability and dynamical properties of conservative and dissipative solitons in settings with spatially inhomogeneous linear refractive index, nonlinearity, gain or absorption, is the subject of this PhD Thesis.
We address stable conservative fundamental and multipole solitons in complex engineered materials with an inhomogeneous linear refractive index and
nonlinearity. We show that stable two‐dimensional solitons may exist in nonlinear lattices with transversally alternating domains with cubic and saturable
nonlinearities. We consider multicomponent solitons in engineered materials, where one field component feels the modulation of the refractive index or
nonlinearity while the other component propagates as in a uniform nonlinear medium. We study whether the cross‐phase‐modulation between two
components allows the stabilization of the whole soliton state.
Media with defocusing nonlinearity growing rapidly from the center to the periphery is another example of a complex engineered material. We study such
systems and, in contrast to the common belief, we have found that stable bright solitons do exist when defocusing nonlinearity grows towards the periphery rapidly enough. We consider different nonlinearity landscapes and analyze the types of soliton solution available in each case.
Nonlinear materials with complex spatial distributions of gain and losses also provide important opportunities for the generation of stable one‐ and
multidimensional fundamental, multipole, and vortex solitons. We study onedimensional solitons in focusing and defocusing nonlinear dissipative materials
with single‐ and double‐well absorption landscapes. In two‐dimensional geometries, stable vortex solitons and complexes of vortices could be observed.
We not only address stationary vortex structures, but also steadily rotating vortex solitons with azimuthally modulated intensity distributions in radially symmetric gain landscapes.
Finally, we study the possibility of forming stable topological light bullets in focusing nonlinear media with inhomogeneous gain landscapes and uniform twophoton absorption
Light Beams in Liquid Crystals
This reprint collects recent articles published on "Light Beams in Liquid Crystals", both research and review contributions, with specific emphasis on liquid crystals in the nematic mesophase. The editors, Prof. Gaetano Assanto (NooEL, University of Rome "Roma Tre") and Prof. Noel F. Smyth (School of Mathematics, University of Edinburgh), are among the most active experts worldwide in nonlinear optics of nematic liquid crystals, particularly reorientational optical solitons ("nematicons") and other all-optical effects
On the Spectral Stability of Solitary Waves
We study the spectral stability of the solitary wave solutions to the nonlinear Dirac equations. We focus on two types of nonlinearity: the Soler type and the Coulomb type. For the Soler model, we apply the Evans function technique to explore the point spectrum of the linearized operator at a solitary wave solution to the 2D and 3D cases.
For the toy Coulomb model, the solitary wave solutions are no longer SU(1, 1) symmetric. We show numerically that there are no eigenvalues near 2ωi in the nonrelativistic limit (ω . m) and the spectral stability persists in spite of the absence of SU(1, 1) symmetry
- …