New features of modulational instability of partially coherent light; importance of the incoherence spectrum

It is shown that the properties of the modulational instability of partially coherent waves propagating in a nonlinear Kerr medium depend crucially on the profile of the incoherent field spectrum. Under certain conditions, the incoherence may even enhance, rather than suppress, the instability. In particular, it is found that the range of modulationally unstable wave numbers does not necessarily decrease monotonously with increasing degree of incoherence and that the modulational instability may still exist even when long wavelength perturbations are stable.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let

Partially incoherent optical vortices in self-focusing nonlinear media

We observe stable propagation of spatially localized single- and double-charge optical vortices in a self-focusing nonlinear medium. The vortices are created by self-trapping of partially incoherent light carrying a phase dislocation, and they are stabilized when the spatial incoherence of light exceeds a certain threshold. We confirm the vortex stabilization effect by numerical simulations and also show that the similar mechanism of stabilization applies to higher-order vortices.Comment: 4 pages and 6 figures (including 3 experimental figures

Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schroedinger equations

We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schroedinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general $N$-coupled nonlinear Schroedinger equations (N-CNLS). It is pointed out that the underlying solitons undergo inelastic (shape changing) collisions due to intensity redistribution among the modes. We also analyse the various possibilities and conditions for such collisions to occur. Further, we report the significant fact that the various partial coherent solitons (PCS) discussed in the literature are special cases of the higher order bright soliton solutions of the N-CNLS equations.Comment: 4 pages, RevTex, 1 EPS figure To appear in Physical Review Letter

Exact soliton solutions of coupled nonlinear Schr\"odinger equations: Shape changing collisions, logic gates and partially coherent solitons

The novel dynamical features underlying soliton interactions in coupled nonlinear Schr{\"o}dinger equations, which model multimode wave propagation under varied physical situations in nonlinear optics, are studied. In this paper, by explicitly constructing multisoliton solutions (upto four-soliton solutions) for two coupled and arbitrary $N$-coupled nonlinear Schr{\"o}dinger equations using the Hirota bilinearization method, we bring out clearly the various features underlying the fascinating shape changing (intensity redistribution) collisions of solitons, including changes in amplitudes, phases and relative separation distances, and the very many possibilities of energy redistributions among the modes of solitons. However in this multisoliton collision process the pair-wise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons. Detailed asymptotic analysis also shows that when solitons undergo multiple collisions, there exists the exciting possibility of shape restoration of atleast one soliton during interactions of more than two solitons represented by three and higher order soliton solutions. From application point of view, we have shown from the asymptotic expressions how the amplitude (intensity) redistribution can be written as a generalized linear fractional transformation for the $N$-component case. Also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters, leading to possible multistate logic. In addition, we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape changing collisions, thereby explaining their variable profile and shape variation in collision process.Comment: 50 Pages, 13 .jpg figures. To appear in PR

Statistical Effects in the Multistream Model for Quantum Plasmas

A statistical multistream description of quantum plasmas is formulated, using the Wigner-Poisson system as dynamical equations. A linear stability analysis of this system is carried out, and it is shown that a Landau-like damping of plane wave perturbations occurs due to the broadening of the background Wigner function that arises as a consequence of statistical variations of the wave function phase. The Landau-like damping is shown to suppress instabilities of the one- and two-stream type.Comment: 5 page

Universality in Systems with Power-Law Memory and Fractional Dynamics

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as "universality", can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201

Stable vortex and dipole vector solitons in a saturable nonlinear medium

We study both analytically and numerically the existence, uniqueness, and stability of vortex and dipole vector solitons in a saturable nonlinear medium in (2+1) dimensions. We construct perturbation series expansions for the vortex and dipole vector solitons near the bifurcation point where the vortex and dipole components are small. We show that both solutions uniquely bifurcate from the same bifurcation point. We also prove that both vortex and dipole vector solitons are linearly stable in the neighborhood of the bifurcation point. Far from the bifurcation point, the family of vortex solitons becomes linearly unstable via oscillatory instabilities, while the family of dipole solitons remains stable in the entire domain of existence. In addition, we show that an unstable vortex soliton breaks up either into a rotating dipole soliton or into two rotating fundamental solitons.Comment: To appear in Phys. Rev.

Fractional Operators, Dirichlet Averages, and Splines

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented

Statistical Theory for Incoherent Light Propagation in Nonlinear Media

A novel statistical approach based on the Wigner transform is proposed for the description of partially incoherent optical wave dynamics in nonlinear media. An evolution equation for the Wigner transform is derived from a nonlinear Schrodinger equation with arbitrary nonlinearity. It is shown that random phase fluctuations of an incoherent plane wave lead to a Landau-like damping effect, which can stabilize the modulational instability. In the limit of the geometrical optics approximation, incoherent, localized, and stationary wave-fields are shown to exist for a wide class of nonlinear media.Comment: 4 pages, REVTeX4. Submitted to Physical Review E. Revised manuscrip

A Multi-Cancer Mesenchymal Transition Gene Expression Signature Is Associated with Prolonged Time to Recurrence in Glioblastoma

A stage-associated gene expression signature of coordinately expressed genes, including the transcription factor Slug (SNAI2) and other epithelial-mesenchymal transition (EMT) markers has been found present in samples from publicly available gene expression datasets in multiple cancer types, including nonepithelial cancers. The expression levels of the co-expressed genes vary in a continuous and coordinate manner across the samples, ranging from absence of expression to strong co-expression of all genes. These data suggest that tumor cells may pass through an EMT-like process of mesenchymal transition to varying degrees. Here we show that, in glioblastoma multiforme (GBM), this signature is associated with time to recurrence following initial treatment. By analyzing data from The Cancer Genome Atlas (TCGA), we found that GBM patients who responded to therapy and had long time to recurrence had low levels of the signature in their tumor samples (P = 3×10−7). We also found that the signature is strongly correlated in gliomas with the putative stem cell marker CD44, and is highly enriched among the differentially expressed genes in glioblastomas vs. lower grade gliomas. Our results suggest that long delay before tumor recurrence is associated with absence of the mesenchymal transition signature, raising the possibility that inhibiting this transition might improve the durability of therapy in glioma patients