Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients

The analytical nonautonomous rogons are reported for the inhomogeneous nonlinear Schrödinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz. These obtained solutions can be used to describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively. Moreover, the snake propagation traces and the fascinating interactions of two nonautonomous rogons are generated for the chosen different parameters. The obtained nonautonomous rogons may excite the possibility of relative experiments and potential applications for the rogue wave phenomenon in the field of nonlinear science.     

Optical vortices in self-focusing Kerr nonlinear media

In this work we numerically compare the interaction of optical vortices (OVs) in self-defocusing and self-focusing Kerr nonlinear media. We find that the basic scenarios (attraction/repulsion, translation/rotation vs. background) in the interaction of two and three vortices with equal and alternative topological charges (TCs) are the same in both media. However, the vortex dynamics under self-focusing conditions is influenced by the reshaping of the surrounding part of the background. Square structure of OVs with alternating TCs is found to be stable with respect to the vortex positions in self-focusing media. This elementary cell is successfully generalized in a large square array of OVs with alternative TCs which brings ordering in the multiple filamentation of the background beam in self-focusing conditions.     

Theory of singular vortex solutions of the nonlinear Schrödinger equation

We present a systematic study of singular vortex solutions of the critical and supercritical two-dimensional nonlinear Schrödinger equation. In particular, we study the critical power for collapse and the asymptotic blowup profile of singular vortices.     

Angular pseudomomentum theory for the generalized nonlinear Schrödinger equation in discrete rotational symmetry media

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schrödinger equation based on the concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schrödinger equation with a nonlinearity depending on the modulus of the field. We provide a rigorous proof of a set of mathematical results justifying that these solitons can be classified according to the irreducible representations of a discrete group. Then we extend this theory to non-stationary solutions and study the relationship between angular momentum and pseudomomentum. We illustrate these theoretical results with numerical examples.     

Hirota method for the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential

In this paper, a Hirota method is developed for applying to the nonlinear Schrödinger equation with an arbitrary time-dependent linear potential which denotes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensation. The nonlinear Schrödinger equation is decoupled to two equations carefully. With a reasonable assumption the one- and two-soliton solutions are constructed analytically in the presence of an arbitrary time-dependent linear potential.     

Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates

We study the properties of the ground state of nonlinear Schrödinger equations with spatially inhomogeneous interactions and show that it experiences a strong localization on the spatial region where the interactions vanish. At the same time, tunneling to regions with positive values of the interactions is strongly suppressed by the nonlinear interactions and as the number of particles is increased it saturates in the region of finite interaction values. The chemical potential has a cutoff value in these systems and thus takes values on a finite interval. The applicability of the phenomenon to Bose-Einstein condensates is discussed in detail.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Internal vibrations of nonlocal nonlinear Schrödinger soliton

We prove that the nonlocality is a source of the internal mode generation in a bright soliton of the nonlocal nonlinear Schrödinger equation, at least in the weak nonlocality limit. The internal mode bifurcates from the edge of the continuous spectrum of the linearized eigenvalue problem into the gap of this spectrum. A dependence of the internal mode propagation constant position in the gap on the nonlocality rate is established. It is shown that the vibration amplitude of the soliton decays inversely proportional to the propagation distance, as in the local models.     

Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Discrete nonlinear Schrödinger equation with arbitrary nonlocality: Linear stability analysis and localized solution

The modulational instability of a plane wave for a discrete nonlinear Schrödinger equation with arbitrary nonlocality is analyzed. This model describes light propagation in a thin film planar waveguide arrays of nematic liquid crystals subjected to a periodic transverse modulation by a low frequency electric field. It is shown that nonlocality can both suppress and promote the growth rate and bandwidth of instability, depending on the type of a response function of a discrete medium. A solitary wave (breather-like) solution is built by the variational approximation and its stability is demonstrated.     

Vortex solitons with inhomogeneous polarization in nonlocal self-focusing nonlinear media

Both azimuthally and radially polarized vortex solitons are investigated to be able to exist in highly nonlocal nonlinear media. We get exactly analytical solutions of azimuthally polarized vortex solitons with only polarization singularities and radially polarized vortex solitons with both phase singularities and polarization singularities. Both azimuthally and radially polarized vortex solitons can exist in nonlocal self-focusing nonlinear media with proper modulation of the beam power and the degree of nonlocality. Contrary to those of radially polarized counterparts in local Kerr media, the topological charge can be any integer. When the topological charge m ≠ 0, both phase singularities and polarization singularities work. When m = 0, the polarization singularities work. Azimuthally polarized vortex solitons with polarization singularities corresponds to the linearly polarized vortex solitons with single charge. Our results show that polarization singularities work the same way as phase singularities in some sense.     

Modulation instability with incoherent white light in self-defocusing photorefractive crystal

We report on the first experimental observation of modulation instability and spontaneous pattern formation with incoherent white light emitted from an incandescent lamp in self-defocusing photorefractive LiNbO₃:Fe crystal. Experimental results show that the modulation instability of white light in self-defocusing medium is related to the input intensity, illumination time and the direction of the crystalline c-axis with respect to that of the lamp filament. At the illumination time t = 0 and the steady-state of MI, we give the spatial distribution of the optical Fourier power spectrum experimentally and the corresponding Fourier transformation of the output intensity numerically, and observe the emergence of the high frequency component along the c-axis.     

Modulation instabilities in a system of four coupled, nonlinear Schrödinger equations

The modulation instability of continuous waves for a system of four coupled nonlinear Schrödinger equations, two of which are in the unstable regime, is studied. In earlier studies, plane or continuous waves for a system of two coupled, nonlinear Schrödinger equations is shown to exhibit modulation instability (MI), even if both modes are in the normal dispersion regime, provided that the coefficient of cross phase modulation (XPM) is larger than that of self phase modulation (SPM). Requirements for MI in this system of four coupled, nonlinear Schrödinger equations can be relaxed. MI can occur even if the magnitude of XPM is less than that of SPM, and the magnitude of instability is generally larger than that of each mode alone. The implications for parametric process and wavelength exchange in optical physics with two pump waves are discussed.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Simultaneous estimation of optical nonlinear refractive and absorptive parameters by solvent induced changes in optical density

With picosecond pulses at the wavelength of 532 nm, we have determined the optical nonlinear refractive and absorptive parameters of the dye N,N′-bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylenedicarboximide (DBPI) by a single closed-aperture (CA) Z-scan technique. This technique uses a theoretical model that elucidates the refractive and absorptive optical nonlinearity present simultaneously in the CA Z-scan profile. The observed remarkable red shift in the absorption spectrum of the dye in the acidic medium as compared to that in the polar medium has been used to vary the optical density at a single frequency. We find that the effect of saturable absorption (SA) is complete at higher concentrations. The effect of reverse saturable absorption (RSA) is dominating in dilute concentrations. The observed variations in the excited state refractive cross-section (σ_r) with the concentration and the energy have been attributed to the contributions of higher order nonlinearity along with the existing third-order nonlinearity.     

Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates

Under investigation in this paper is a nonlinear Schrödinger equation with an arbitrary linear time-dependent potential, which governs the soliton dynamics in quasi-one-dimensional Bose-Einstein condensates (quasi-1DBECs). With Painlevé analysis method performed to this model, its integrability is firstly examined. Then, the distinct treatments based on the truncated Painlevé expansion, respectively, give the bilinear form and the Painlevé-Bäcklund transformation with a family of new exact solutions. Furthermore, via the computerized symbolic computation, a direct method is employed to easily and directly derive the exact analytical dark- and bright-solitonic solutions. At last, of physical and experimental interests, these solutions are graphically discussed so as to better understand the soliton dynamics in quasi-1DBECs.     

Effects of metallization and bromination on nonlinear optical properties of diphenylporphyrins

Nonlinear refraction and nonlinear absorption of diphenylporphyrins with bromination and metallization were studied by Z-scan technique in nanosecond and picosecond regimes. Results show that both metallization and bromination of diphenylporphyrins can cause the regular change of magnitude and sign of nonlinear absorption. The transition between saturable absorption and reverse saturable absorption happens as bromine increases and metal ion changes. The effect of bromination on nonlinear refraction is small, the change of nonlinear refraction is mainly attributed to the metallization. Influences of photophysical parameters on nonlinear absorption and nonlinear refraction have been elucidated using five-level models.     

Non-quantum liquefaction of coherent gases

We show that a gas-to-liquid phase transition at zero temperature may occur in a coherent gas of bosons in the presence of competing nonlinear effects. This situation can take place in atomic systems like Bose-Einstein condensates in alkali gases with two-body and three-body interactions of opposite signs, as well as in laser beams which propagate through optical media with Kerr (focusing) and higher order (defocusing) nonlinear responses. The liquefaction process takes place in the absence of any quantum effect and can be formulated in the framework of a mean field theory, in terms of the minimization of a thermodynamic potential. We study from a thermodynamic point of view all the stationary solutions of the cubic-quintic nonlinear Schrödinger equation which describes our system. We show that solitonic localized solutions connect the gaseous and liquid phases. Furthermore, we also perform a numerical simulation in the presence of linear gain and three-body recombination where a rich dynamics, including the emergence of self-organization behavior, is found.     

Modulational Instability of （1 ＋ 1）-Dimensional Bose-Einstein Condensate with Three-Body Interatomic Interaction

The modulational instability of Bose-Einstein condensate with three-body interatomic interaction and external harmonic trapping potential is investigated. Both of our analytical and numerical results show that the external potential will either cause the excitation of modulationally unstable modes or restrain the modulationally unstable modes from growing.     

High sensitive measurements of absorption coefficient and optical nonlinearities

Accurate knowledge of absorption coefficient of a sample is a prerequisite for measuring the third-order optical nonlinearity of materials, which can be a serious limitation for unknown samples. We introduce a method, which measures both the absorption coefficient and the third-order optical nonlinearity of materials with high sensitivity in a single experimental arrangement. We use a dual-beam pump-probe experiment and conventional single-beam z-scan under different conditions to achieve this goal with communication relevant wavelength. We also demonstrate a counterintuitive coupling of the non-interacting probe-beam with the pump-beam in pump-probe z-scan experiment.