Elliptic Solitons Research Articles

Ermakov–Ray–Reid Reductions of Variational Approximations in Nonlinear Optics

Integrable Hamiltonian systems of Ermakov‐Ray‐Reid type are shown to arise out of variational approximation to certain modulated NLS models as well as in spiralling elliptic soliton systems and their generalization in a Bose‐Einstein setting.

Elliptical Gaussian solitons in synthetic nonlocal nonliear media

This paper studies analytically and numerically the dynamics of two-dimensional elliptical Gaussian solitons in a “double-self-focusing" synthetic nonlocal media featuring elliptical and circular Gaussian response with different degrees of nonlocality. Based on the variational approach, it obtains the approximately analytical solution of such Gaussian elliptical solitons. It also computes the stability of the solitons by numerical simulations.

Nonlinear phenomena, optical and quantum solitons

Research in the theory of solitons and its applications has undergone impressive development in recent years and has affected a broad range of fields, including theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups and

Dipole solitons in nonlocal nonlinear media with anisotropy

We investigate the existence and stability of dipole-mode solitons in two-dimensional models of nonlocal media with anisotropic Kerr nonlinearity analytically and numerically. We obtain the approximate solution of such elliptic dipole solitons by using the variational approximation. The dynamics of such dipole-mode solitons is governed by the eccentricity of both the input beam and the nonlocal response function. We also compute the stability of the solitons by direct numerical simulations. The effects of the anisotropy of the nonlocal response function on the propagation of the dipole beam are also discussed in detail.

Two-dimensional gap solitons in elliptic-lattice potentials

We study two-dimensional (2D) matter-wave gap solitons trapped in an elliptically deformed concentric lattice potential, within the framework of the Gross-Pitaevskii equation (GPE) with self-attraction or self-repulsion. For a fixed eccentricity of the lattice, soliton families are found in both the repulsive and attractive models. In the former case, the analysis reveals two kinds of gap solitons trapped in the first oval trough (the ring-shaped potential minimum closest to the center): elliptic annular solitons (EASs), and double solitons (DSs), which are formed by two tightly localized density peaks located at diametrically opposite points of the trough, with zero phase difference between them. With the decrease of the norm, the density distribution in the EAS along the azimuthal direction changes from nearly uniform to double-peaked and, eventually, to the DS. In the attractive model, there exist only DSs in the oval trough, while EASs are not found. All such solitons without the angular momentum ($l=0$) are fully stable. For $l\ensuremath{\ne}0$, vortical solitons---both EASs with a sufficiently large norm (in the repulsive model) and DSs (in models with both signs of the nonlinearity)---are quasistable, exhibiting rocking motion in the elliptic trough (we consider the cases of $l=1 \mathrm{and} l=2$). At smaller values of the norm, the vortical annular solitons (in the repulsive model) are unstable. Stable fundamental solitons trapped in the central potential well are investigated, too, in both the attractive and repulsive models, by means of the variational approximation and numerical methods.

Symbolic computation of exact solutions for the compound KdV-Sawada–Kotera equation

The generalized F-expansion method is applied to construct the exact solutions of the compound KdV-Sawada–Kotera equation by the aid of the symbolic computation system Maple. Some new exact solutions which include Jacobi elliptic function solutions, soliton solutions and triangular periodic solutions are obtained via this method.

Jacobian elliptic solitons in inhomogeneous alpha-helical proteins

In this paper, we study the effects of mass inhomogeneities on the energy transfer by solitons in alpha-helical proteins. We demonstrate that such a system is governed by a standard elliptic type equation. Total energy of the soliton and its effective mass are calculated. We determine critical solitons' velocities at the borders of the inhomogeneous zone. In the case of light-mass-induced impurities, we obtain that oscillatory motions are possible provided cn-type solitons are considered. In the case of heavy-mass-induced impurities, oscillatory motion can be introduced by dn-type solitons. Pinning frequencies of oscillations are estimated.

The linear and nonlinear optical effects of white light

An overview of our research group’s experimental and theoretical developments is provided on the linear and nonlinear optical effects of white light since 2003. Their work includes the experimental researches on the white light one-dimensional photovoltaic dark spatial solitons and the waveguides and directional couplers induced by them, the circular and elliptic white-light dark spatial solitons and the white-light photorefractive phase masks, two-dimensional white-light photonic lattices and the applications of the white-light dark spatial solitons in the digital image transmission field, the interaction between the two-dimensional white-light dark spatial solitons to enhance or to improve the correlated degree of the white light through the interaction between the white-light beam and coherent dark spatial solitons, the interaction between the one- or two-dimensional white-light dark spatial solitons and the two-dimensional white-light photonic lattices, respectively. We also numerically simulate the interaction between two or more partially incoherent bright spatial solitons and the white bright spatial soliton pairs in the saturated logarithmic nonlinear medium. We have observed experimentally for the first time, the modulation instability of the coherent light and white light, respectively, in self-defocusing medium and so on.

Symmetry-breaking instabilities of generalized elliptical solitons

Elliptical solitons in 2D nonlinear Schödinger equations are studied numerically with a more-generalized formulation. New families of solitons, vortices, and soliton rings with elliptical symmetry are found and investigated. With a suitable symmetry-breaking parameter, we show that perturbed elliptical solitons tend to move transversely owing to the existences of dipole excitation modes, which are totally suppressed in circularly symmetric solitons. Furthermore, by numerical evolutions we demonstrate that elliptical vortices and soliton rings collapse into a pair of stripes and clusters, respectively, revealing the experimental observations in the literature.

Elliptic incoherent spatial solitons and their interactions in strongly nonlocal media with an anisotropic nonlocality

We investigate the propagation and interaction of elliptic incoherent spatial solitons (EISS) in strongly nonlocal kerr media with an anisotropic nonlocality based on the coherent density approach. An exact analytical solution of such EISS is obtained; the results show that such EISS can form with both isotropic and anisotropic coherence. Moreover, we find that the interaction properties of EISS are very similar to that of their coherent counterpart. Some numerical examples are presented and pertinent physics features are addressed.

Elliptical discrete solitons supported by enhanced photorefractive anisotropy

We demonstrate elliptical discrete solitons in an optically induced two-dimensional photonic lattice. The ellipticity of the discrete soliton results from enhanced photorefractive anisotropy and nonlocality under a nonconventional bias condition. We show that the ellipticity and orientation of the discrete solitons can be altered by changing the direction of the lattice beam and/or the bias field relative to the crystalline c-axis. Our experimental results are in good agreement with the theoretical prediction.

Elliptic incoherent solitons in strongly nonlocal media with anisotropic Kerr nonlinearity

The propagation of elliptic incoherent beam in strongly nonlocal media with noninstantaneous anisotropic Kerr nonlinearity is systematically investigated. Using the mutual coherence function approach, we obtain the existence curve of such solitons and an interesting outcome: correlation characteristics of the elliptic incoherent solitons can be anisotropic as well as isotropic. When the existence conditions of solitons are not satisfied, the elliptic incoherent beam will undergo periodic oscillation. Nonstationary evolution behaviors of the elliptic beam are shown in detail by numerical calculation. It is also obtained that the oscillation periods of the beam in x and y direction are different and the elliptic beam will become circular at some propagation distance under special conditions.

Elliptical solitons in nonconventionally biased photorefractive crystals

We theoretically predict and experimentally observe the two-dimensional (2-D) bright solitons in a nonconventionally biased strontium barium niobate (SBN) crystal. A theory describing light propagating in an SBN crystal with a bias field along an arbitrary direction is formulated. Then the existence of 2-D bright solitons in such a crystal is numerically verified. By employing digital holography, the index changes induced by Gaussian beams in an SBN crystal under different biasing conditions are visualized. Finally, skewed elliptical solitons are experimentally demonstrated.

Elliptic incoherent accessible solitons in strongly nonlocal media

We study the propagation of elliptic incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. For this soliton to exist, the coherence properties of the incoherent beam should be anisotropic. The total power of the incident beam should also equal to a critical value which depend on the beam width as well as the coherence properties. When initial parameters of the beam do not satisfy the existence conditions, the elliptic incoherent accessible solitons will undergo linear harmonic oscillation in different states. Corresponding properties are studied in detail.

Solitons in Nonlinear Media with an Infinite Range of Nonlocality: First Observation of Coherent Elliptic Solitons and of Vortex-Ring Solitons

We present an experimental study on wave propagation in highly nonlocal optically nonlinear media, for which far-away boundary conditions significantly affect the evolution of localized beams. As an example, we set the boundary conditions to be anisotropic and demonstrate the first experimental observation of coherent elliptic solitons. Furthermore, exploiting the natural ability of such nonlinearities to eliminate azimuthal instabilities, we perform the first observation of stable vortex-ring solitons. These features of highly nonlocal nonlinearities affected by far-away boundary conditions open new directions in nonlinear science by facilitating remote control over soliton propagation.

The circular and elliptic white-light dark spatial solitons in a photovoltaic self-defocusing nonlinear medium

We observed circular white-light dark spatial solitons in photovoltaic self-defocusing LiNbO3:Fe crystal using an ordinary incandescent lamp as light source (a line source) with the filament of the bulb parallel to the crystalline c axis. Besides the above condition, the formation of the elliptic soliton needs an additive condition that the crystalline c axis is parallel to the minor axis of dark elliptic spot.

Exact solutions for the coupled Klein–Gordon–Schrödinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled Klein-Gordon Schrodinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

Exact Localized and Periodic Solutions of the Ablowitz-Ladik Discrete Nonlinear Schrödinger System

We have studied, analytically, the Ablowitz-Ladik discrete nonlinear Schr¨odinger system. We have found a set of exact solutions which includes as particular cases periodic solutions in terms of elliptic Jacobian functions, bright and dark soliton solutions, and quasi-periodic solutions. We have also found the range of parameters where each exact solution exists. - PACS: 02.30.Jr, 05.45.Yv, 42.65.Tg, 02.30.Gp.

Nonlinear dual-core photonic crystal fiber couplers

We study nonlinear modes of dual-core photonic crystal fiber couplers made of a material with the focusing Kerr nonlinearity. We find numerically the profiles of symmetric, antisymmetric, and asymmetric nonlinear modes and analyze all-optical switching generated by the instability of the symmetric mode. We also describe elliptic spatial solitons controlled by the waveguide boundaries.

A Connection Between Geometry and Dynamical Systems

This paper deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the anisotropic harmonic oscillator in a radial quartic potential. We study this system from a different angle: a) We show, using a Lax-type representation of the Hamilton’s equations of motion, that the system is linearized in the jacobian variety of a smooth genus 2 hyperelliptic curve. b) We find via Kowalewski-Painlevé analysis the principal balances of the hamiltonian vector field defined by the hamiltonian and we show that the system is algebraic complete integrable. c) We also describe an explicit embedding of the abelian variety which completes the generic invariant surface, into projective space. d) We give a direct proof that the abelian variety obtained in this paper is dual to Prym variety and can also be seen as a double unramified cover of the jacobian variety of an hyperelliptic curve of genus 2. e) We show that at some special values of the parameters λ1 and λ2, we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation.