Search results for " Pattern formation"
showing 10 items of 73 documents
Similarity Solutions and Collapse in the Attractive Gross-Pitaevskii Equation
2000
We analyse a generalised Gross-Pitaevskii equation involving a paraboloidal trap potential in $D$ space dimensions and generalised to a nonlinearity of order $2n+1$. For {\em attractive} coupling constants collapse of the particle density occurs for $Dn\ge 2$ and typically to a $\delta$-function centered at the origin of the trap. By introducing a new dynamical variable for the spherically symmetric solutions we show that all such solutions are self-similar close to the center of the trap. Exact self-similar solutions occur if, and only if, $Dn=2$, and for this case of $Dn=2$ we exhibit an exact but rather special D=1 analytical self-similar solution collapsing to a $\delta$-function which …
Diffusion stabilizes cavity solitons in bidirectional lasers
2009
We study the influence of field diffusion on the spatial localized structures (cavity solitons) recently predicted in bidirectional lasers. We find twofold positive role of the diffusion: 1) it increases the stability range of the individual (isolated) solitons; 2) it reduces the long-range interaction between the cavity solitons. Latter allows the independent manipulation (writing and erasing) of individual cavity solitons.
Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding
2016
We introduce $\mathcal{B}_{\kappa}$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $\kappa$, a finite or denumerable set of objects at $\kappa=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $\kappa \to \infty$. We show that $\mathcal{B}_{\kappa}$-embeddings can be used to design nonlinear irreversible processes through this connection. A number of examples of increasing complexity are worked out to illustrate the possibilities uncovered by this concept. These include not only smooth functions but also fractals on the real line and on the complex plane. As an application, we use $\mat…
Lieb polariton topological insulators
2018
We predict that the interplay between the spin-orbit coupling, stemming from the TE-TM energy splitting, and the Zeeman effect in semiconductor microcavities supporting exci- ton-polariton quasi-particles results in the appearance of unidirectional linear topological edge states when the top microcavity mirror is patterned to form a truncated dislocated Lieb lattice of cylindrical pillars. Periodic nonlinear edge states are found to emerge from the linear ones. They are strongly localized across the interface and they are remarkably robust in comparison to their counterparts in hexagonal lattices. Such robustness makes possible the existence of nested unidirectional dark solitons that move …
Faraday patterns in bose-Einstein condensates.
2002
Temporal periodic modulation of the interatomic s-wave scattering length in Bose-Einstein condensates is shown to excite subharmonic patterns of atom density through a parametric resonance. The dominating wavelength of the spatial structures is shown to be primarily selected by the excitation frequency but also affected by the depth of the spatial modulation via a nonlinear resonance. These phenomena represent macroscopic quantum analogues of the Faraday waves excited in vertically shaken liquids.
Turing Instability and Pattern Formation in an Activator-Inhibitor System with Nonlinear Diffusion
2014
In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; %favors the mechanism of pattern formation with respect to the classical linear diffusion case; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude o…
Phase Domains and Spatial Solitons in Degenerate Optical Parametric Oscillators with Injection
2000
The stability of phase domains and spatial solitons in DOPO under the presence of an injected signal is investigated. The injected signal prevents the nondegenerate regime and, for a particular value of the phase, preserves the equivalence between the two homogeneous states, allowing the domain formation and, in particular, the stability of solitons. The main conclusion is that injection facilitates the experimental observation of solitons in degenerate OPOs.
Pattern formation in a complex Swift-Hohenberg equation with phase bistability
2016
We study pattern formation in a complex Swift Hohenberg equation with phase-sensitive (parametric) gain. Such an equation serves as a universal order parameter equation describing the onset of spontaneous oscillations in extended systems submitted to a kind of forcing dubbed rocking when the instability is towards long wavelengths. Applications include two-level lasers and photorefractive oscillators. Under rocking, the original continuous phase symmetry of the system is replaced by a discrete one, so that phase bistability emerges. This leads to the spontaneous formation of phase-locked spatial structures like phase domains and dark-ring (phase-) cavity solitons. Stability of the homogeneo…
Asymptotic regime in N random interacting species
2005
The asymptotic regime of a complex ecosystem with \emph{N}random interacting species and in the presence of an external multiplicative noise is analyzed. We find the role of the external noise on the long time probability distribution of the i-th density species, the extinction of species and the local field acting on the i-th population. We analyze in detail the transient dynamics of this field and the cavity field, which is the field acting on the $i^{th}$ species when this is absent. We find that the presence or the absence of some population give different asymptotic distributions of these fields.
MEAN FIELD APPROACH AND ROLE OF THE COLOURED NOISE IN THE DYNAMICS OF THREE INTERACTING SPECIES
2010
We study the effects of the coloured noise on the dynamics of three interacting species, namely two preys and one predator, in a two-dimensional lattice with N sites. The three species are affected by multiplicative time correlated noise, which accounts for the effects of environment on the species evolution. Moreover, the interaction parameter between the two preys is a dichotomous stochastic process, which determines two dynamical regimes corresponding to different biological conditions. Preliminarily, we study the noise effect on the three species dynamics in single site. Then, we use a mean field approach to obtain, in Gaussian approximation, the moment equations for the species densiti…