Collapse in the nonlocal nonlinear Schr\"odinger equation

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both, Lyapunoff's method and virial identities. We find that for for a one-dimensional case, i.e. for $n=1$, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension $n\geq2$ and singular kernel $\sim 1/r^\alpha$, no collapse takes place if $\alpha<2$, whereas collapse is possible if $\alpha\ge2$. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of $\sim 1/r^2$ kernels. Moreover, different evolution scenarios for the three dimensional physically relevant case of Bose Einstein condensate are studied numerically for both, the ground state and a higher order toroidal state with and without an additional local repulsive nonlinear interaction. In particular, we show that presence of an additional local repulsive term can prevent collapse in those cases

Pattern formation in the nonlinear Schrödinger equation with competing nonlocal nonlinearities

We study beam propagation in the framework of the nonlinear Schrödinger equation with competing Gaussian nonlocal nonlinearities. We demonstrate that such system can give rise to self-organization of light into stable states of trains or hexagonal arrays of filaments, depending on the transverse dimensionality. This long-range ordering can be achieved by mere unidirectional beam propagation. We discuss the dynamics of long-range ordering and the crucial role which the phase of the wavefunction plays for this phenomenon. Furthermore we discuss how transverse dimensionality affects the order of the phasetransition

Stability of two-dimensional spatial solitons in nonlocal nonlinear media

We discuss existence and stability of two-dimensional solitons in media with spatially nonlocal nonlinear response. We show that such systems, which include thermal nonlinearity and dipolar Bose Einstein condensates, may support a variety of stationary localized structures - including rotating spatial solitons. We also demonstrate that the stability of these structures critically depends on the spatial profile of the nonlocal response function.Comment: 8 pages, 9 figure

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

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

Observation of multivortex solitons in photonic lattices

We report on the first observation of topologically stable spatially localized multivortex solitons generated in optically induced hexagonal photonic lattices. We demonstrate that topological stabilization of such nonlinear localized states can be achieved through self-trapping of truncated two-dimensional Bloch waves and confirm our experimental results by numerical simulations of the beam propagation in weakly deformed lattice potentials in anisotropic photorefractive media

Applications of Two-Body Dirac Equations to the Meson Spectrum with Three versus Two Covariant Interactions, SU(3) Mixing, and Comparison to a Quasipotential Approach

In a previous paper Crater and Van Alstine applied the Two Body Dirac equations of constraint dynamics to the meson quark-antiquark bound states using a relativistic extention of the Adler-Piran potential and compared their spectral results to those from other approaches, ones which also considered meson spectroscopy as a whole and not in parts. In this paper we explore in more detail the differences and similarities in an important subset of those approaches, the quasipotential approach. In the earlier paper, the transformation properties of the quark-antiquark potentials were limited to a scalar and an electromagnetic-like four vector, with the former accounting for the confining aspects of the overall potential, and the latter the short range portion. A part of that work consisted of developing a way in which the static Adler-Piran potential was apportioned between those two different types of potentials in addition to covariantization. Here we make a change in this apportionment that leads to a substantial improvement in the resultant spectroscopy by including a time-like confining vector potential over and above the scalar confining one and the electromagnetic-like vector potential. Our fit includes 19 more mesons than the earlier results and we modify the scalar portion of the potential in such a way that allows this formalism to account for the isoscalar mesons {\eta} and {\eta}' not included in the previous work. Continuing the comparisons made in the previous paper with other approaches to meson spectroscopy we examine in this paper the quasipotential approach of Ebert, Faustov, and Galkin for a comparison with our formalism and spectral results.Comment: Revisions of earlier versio

Soliton molecules in trapped vector Nonlinear Schrodinger systems

We study a new class of vector solitons in trapped Nonlinear Schrodinger systems modelling the dynamics of coupled light beams in GRIN Kerr media and atomic mixtures in Bose-Einstein condensates. These solitons exist for different spatial dimensions, their existence is studied by means of a systematic mathematical technique and the analysis is made for inhomogeneous media

Tracking azimuthons in nonlocal nonlinear media

We study the formation of azimuthons, i.e., rotating spatial solitons, in media with nonlocal focusing nonlinearity. We show that whole families of these solutions can be found by considering internal modes of classical non-rotating stationary solutions, namely vortex solitons. This offers an exhaustive method to identify azimuthons in a given nonlocal medium. We demonstrate formation of azimuthons of different vorticities and explain their properties by considering the strongly nonlocal limit of accessible solitons.Comment: 11 pages, 7 figure

Collapse arrest and soliton stabilization in nonlocal nonlinear media

We investigate the properties of localized waves in systems governed by nonlocal nonlinear Schrodinger type equations. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlinearity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr media in all physical dimensions. The nonlocal nonlinear response must be symmetric, but can be of completely arbitrary shape. We use variational techniques to find the soliton solutions and illustrate the stabilizing effect of nonlocality.Comment: 4 pages with 3 figure