Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential

Stability of trapped Bose-Einstein condensate under a density-dependent gauge field

We study the ground-state stability of the trapped one-dimensional Bose-Einstein condensate under a density-dependent gauge field by variational and numerical methods. The competition of density-dependent gauge field and mean-field atomic interaction induces the instability of the ground state, which results in irregular dynamics. The threshold of the gauge field for exciting the instability is obtained analytically and confirmed numerically. When the gauge field is less than the threshold, the system is stable, and the gauge field induces chiral dynamics of the wave packet. When the gauge field is greater than the threshold, the system is unstable, and the ground-state wave packet will be deformed and fragmented. Interestingly, we find that as the gauge field approaches the threshold, strong dipolar and breathing dynamics take place, and strong modes mixing occurs, the instability of the system sets in. In addition, we show that the stability of the system can be well controlled by periodical modulation of the trapping potential. We provide theoretical evidence to understand and control the irregular dynamics associated with chiral superfluid induced by density-dependent gauge field.

Some Inequalities Related to Jensen-Type Results with Applications

The class of harmonic convex functions has acquired a very useful and significant placement among the non-convex functions, since this class not only reinforces some major results of the class of convex functions, but also has supported the development of some remarkable results in analysis where the class of convex functions is silent. Therefore, many researchers have deployed themselves to explore valuable results for this class of non-convex functions. This paper obtains new discrete inequalities for univariate harmonic convex functions on linear spaces related to a Jensen-type and a variant of the Jensen-type results. Our results are refinements of very important recent inequalities presented by Dragomir and Baloch et al. Furthermore, we provide the natural applications of our results.

On Special Properties for Continuous Convex Operators and Related Linear Operators

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

Stability limits for modes held in alternating trapping-expulsive potentials

We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the backdrop of the critical collapse driven by the cubic self-attraction with strength g. The TEM scheme may be implemented, as spatially or temporally periodic modulations, in optics or BEC, respectively. The consideration is carried out by dint of numerical simulations and variational approximation (VA). In terms of the VA, the dynamics amounts to a nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu equation. Stability boundaries are found as functions of g and parameters of the periodic modulation of the trapping potential. Below the usual collapse threshold, which is known, in the numerical form, as g<g_{c}^{(num)}≈5.85 (in the standard notation), the stability is limited by the onset of the parametric resonance. This stability limit, including the setup with the self-repulsive sign of the cubic term (g<0), is accurately predicted by the VA. At g>g_{c}^{(num)}, the collapse threshold is found with the help of full numerical simulations. The relative increase of g_{c} above g_{c}^{(num)} is ≈1.5%. It is a meaningful result, even if its size is small, because the collapse threshold is a universal constant which is difficult to change.

Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein–Gordon Equation

This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.

Contribution of Using Hadamard Fractional Integral Operator via Mellin Integral Transform for Solving Certain Fractional Kinetic Matrix Equations

Recently, the importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in electrodynamics, control systems, economic, physics, geophysics and hydrodynamics. Among the many fractional differential equations are kinetic equations. Fractional-order kinetic Equations (FOKEs) are a unifying tool for the description of load vector behavior in disorderly media. In this article, we employ the Hadamard fractional integral operator via Mellin integral transform to establish the generalization of some fractional-order kinetic equations including extended (k,τ)-Gauss hypergeometric matrix functions. Solutions to certain fractional-order kinetic matrix Equations (FOKMEs) involving extended (k,τ)-Gauss hypergeometric matrix functions are also introduced. Moreover, several special cases of our main results are archived.

Two-Dimensional Solitons in Bose–Einstein Condensates with Spin–Orbit Coupling and Rydberg–Rydberg Interaction

Applying an imaginary time evolution method (AITEM) to the system of Gross–Pitaevskii equations, we find two-dimensional stable solitons in binary atomic Bose–Einstein condensates with spin–orbit coupling (SOC) and the Rydberg–Rydberg interaction (RRI). The stability of 2D solitons by utilizing their norm and energy is discussed in detail. Depending on the SOC and Rydberg–Rydberg interaction, we find stable zero-vorticity and vortical solitons. Furthermore, we show that the solitons can be effectively tuned by the local and nonlocal nonlinearities of this system.

Trapping wave fields in an expulsive potential by means of linear coupling

We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.

A genetic algorithm and backpropagation neural network based temperature compensation method of spin-exchange relaxation-free co-magnetometer

This paper presents a temperature compensation method based on the genetic algorithm (GA) and backpropagation (BP) neural network to reduce the temperature induced error of the spin-exchange relaxation-free (SERF) co-magnetometer. The fluctuation of the cell temperature results in the variation of the optical rotation angle and the probe light absorption. The temperature fluctuation of the magnetic field shielding layer induces the variation of the magnetic field. In addition, one of the causes of light power variation is temperature fluctuation of the optical element. In summary, temperature fluctuations cause a variety of SERF co-magnetometer errors, and the relationship between these errors and temperature fluctuations has the characteristics of time-variance and non-linearity. There are two kinds of methods to suppress these errors. One way is to reduce temperature fluctuations of the SERF co-magnetometer. However, this method requires additional hardware and high cost, which are not suitable for miniaturization and low cost applications. Another effective method to suppress nonlinear and time-varying errors is to utilize intelligent algorithms for temperature compensation. In this paper, the BP neural network is applied for temperature compensation, and the GA is utilized to overcome the disadvantages of the BP neural network. The training data were obtained by changing the ambient temperature of the SERF co-magnetometer. The experimental results show that the method proposed in this work can significantly improve the accuracy of the co-magnetometer at complex ambient temperatures, and the stability of the SERF co-magnetometer at room temperature can be improved by at least 45%.

Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons

We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.

Localization and spin dynamics of spin-orbit-coupled Bose-Einstein condensates in deep optical lattices

We analytically and numerically discuss the dynamics of two pseudospin components Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in deep optical lattices. Rich localized phenomena, such as breathers, solitons, self-trapping, and diffusion, are revealed and strongly depend on the strength of the atomic interaction, SOC, Raman detuning, and the spin polarization (i.e., the initial population difference of atoms between the two pseudospin components of BECs). The critical conditions for the transition of localized states are derived analytically. Based on the critical conditions, the detailed dynamical phase diagram describing the different dynamical regimes is derived. When the Raman detuning satisfies a critical condition, localized states with a fixed initial spin polarization can be observed. When the critical condition is not satisfied, we use two quenching methods, i.e., suddenly and linearly quenching Raman detuning from the soliton or breather state, to discuss the spin dynamics, phase transition, and wave packet dynamics by numerical simulation. The sudden quenching results in a damped oscillation of spin polarization and transforms the system to a new polarized state. Interestingly, the linear quenching of Raman detuning induces a controllable phase transition from an unpolarized phase to an expected polarized phase, while the soliton or breather dynamics is maintained.

A Bose–Einstein condensate is a Bose condensate in the laboratory ground state

Bose–Einstein condensates of weakly interacting, ultra-cold atoms have become a workhorse for exploring quantum effects on atomic motion, but does this condensate need to be in the ground state of the system? Researchers often perform transformations so that their Hamiltonians are easier to analyse. However, changing Hamiltonians can require an energy shift. We show that transforming into a rotating or oscillating frame of reference of a Bose condensate does not then satisfy Einstein’s requirement that a condensate exists in the zero kinetic energy state. We show that Bose condensation can occur above the ground state and at room temperature, referring to recent literature.

Magnification inferred curvature for real-time curvature monitoring

The in situ and real-time measurement of curvature changes of optically reflecting surfaces is a key element to better control bottom-up fabrication processes in the semiconductor industry, but also to follow or adjust mirror deformations during fabrication and use for space or optics industries. Despite progresses made in the last two decades thanks to laser deflectometry-based techniques, the community lacks an instrument, easy to use, robust to tough environments and easily compatible with a large range of fabrication processes. We describe here a new method, called magnification inferred curvature (MIC), based on the determination of the magnification factor of the virtual image size of a known object created by a reflecting curved surface (the substrate) acting as a spherical mirror. The optical formalism, design, and proof of concept are presented. The precision, accuracy, and advantages of the MIC method are illustrated from selected examples taken from real-time growth monitoring and compared with state-of-the-art laser deflectometry-based instruments.

Unsteady thermal Maxwell power law nanofluid flow subject to forced thermal Marangoni Convection

In the current work, the unsteady thermal flow of Maxwell power-law nanofluid with Welan gum solution on a stretching surface has been considered. The flow is also exposed to Joule heating and magnetic effects. The Marangoni convection equation is also proposed for current investigation in light of the constitutive equations for the Maxwell power law model. For non-dimensionalization, a group of similar variables has been employed to obtain a set of ordinary differential equations. This set of dimensionless equations is then solved with the help of the homotopy analysis method (HAM). It has been established in this work that, the effects of momentum relaxation time upon the thickness of the film is quite obvious in comparison to heat relaxation time. It is also noticed in this work that improvement in the Marangoni convection process leads to a decline in the thickness of the fluid’s film.

Testing short distance anisotropy in space

The isotropy of space is not a logical requirement but rather is an empirical question; indeed there is suggestive evidence that universe might be anisotropic. A plausible source of these anisotropies could be quantum gravity corrections. If these corrections happen to be between the electroweak scale and the Planck scale, then these anisotropies can have measurable consequences at short distances and their effects can be measured using ultra sensitive condensed matter systems. We investigate how such anisotropic quantum gravity corrections modify low energy physics through an anisotropic deformation of the Heisenberg algebra. We discuss how such anisotropies might be observed using a scanning tunnelling microscope.

Entropy generation and dissipative heat transfer analysis of mixed convective hydromagnetic flow of a Casson nanofluid with thermal radiation and Hall current

The present article provides a detailed analysis of entropy generation on the unsteady three-dimensional incompressible and electrically conducting magnetohydrodynamic flow of a Casson nanofluid under the influence of mixed convection, radiation, viscous dissipation, Brownian motion, Ohmic heating, thermophoresis and heat generation. At first, similarity transformation is used to transform the governing nonlinear coupled partial differential equations into nonlinear coupled ordinary differential equations, and then the resulting highly nonlinear coupled ordinary differential equations are numerically solved by the utilization of spectral quasi-linearization method. Moreover, the effects of pertinent flow parameters on velocity distribution, temperature distribution, concentration distribution, entropy generation and Bejan number are depicted prominently through various graphs and tables. It can be analyzed from the graphs that the Casson parameter acts as an assisting parameter towards the temperature distribution in the absence of viscous and Joule dissipations, while it has an adverse effect on temperature under the impacts of viscous and Joule dissipations. On the contrary, entropy generation increases significantly for larger Brinkman number, diffusive variable and concentration ratio parameter, whereas the reverse effects of these parameters on Bejan number are examined. Apart from this, the numerical values of some physical quantities such as skin friction coefficients in x and z directions, local Nusselt number and Sherwood number for the variation of the values of pertinent parameters are displayed in tabular forms. A quadratic multiple regression analysis for these physical quantities has also been carried out to improve the present model's effectiveness in various industrial and engineering areas. Furthermore, an appropriate agreement is obtained on comparing the present results with previously published results.

Ground-state phase and superfluidity of tunable spin-orbit-coupled Bose-Einstein condensates

We theoretically study the ground-state phases and superfluidity of tunable spin-orbit-coupled Bose-Einstein condensates (BECs) under the periodic driving of Raman coupling. An effective time-independent Floquet Hamiltonian is proposed by using a high-frequency approximation, and we find single-particle dispersion, spin-orbit-coupling, and asymmetrical nonlinear two-body interaction can be modulated effectively by the periodic driving. The critical Raman coupling characterizing the phase transition and relevant physical quantities in three different phases (the stripe phase, plane-wave phase, and zero momentum phase) are obtained analytically. Our results indicate that the boundary of ground-state phases can be controlled and the system will undergo three different phase transitions by adjusting the external driving. Interestingly, we find the contrast of the stripe density can be enhanced by the periodic driving in the stripe phase. We also study the superfluidity of tunable spin-orbit-coupled BECs and find the dynamical instability can be tuned by the periodic driving of Raman coupling. Furthermore, the sound velocity of the ground-state and superfluidity state can be controlled effectively by tuning the periodic driving strength. Our results indicate that the periodic driving of Raman coupling provides a powerful tool to manipulate the ground-state phase transition and dynamical instability of spin-orbit-coupled BECs.

Perturbation theory for Bose–Einstein condensates on bounded space domains

Bose–Einstein condensates (BECs), first predicted theoretically by Bose and Einstein and finally discovered experimentally in the 1990s, continue to motivate theoretical and experimental physics work. Although experiments on BECs are carried out in bounded space domains, theoretical work in the modelling of BECs often involves solving the Gross–Pitaevskii equation on unbounded domains, as the combination of bounded domains and spatial heterogeneity render most existing analytical approaches ineffective. Motivated by a lack of theory for BECs on bounded domains, we first derive a perturbation theory for both ground and excited stationary states on a given bounded space domain, allowing us to explore the role various forms of the self-interaction, external potential and space domain have on BECs. We are able to show that the shape and curvature of a space domain strongly influence BEC structure, and may be used as control mechanisms in experiments. We next derive a non-autonomous perturbation theory to predict BEC response to temporal changes in an external potential. In certain cases, our approach can be extended to unbounded domains, and we conclude by constructing a perturbation theory for bright solitons within external potentials on unbounded domains.

Heat transfer flow of Maxwell hybrid nanofluids due to pressure gradient into rectangular region

In this work, influence of hybrid nanofluids (Cu and [Formula: see text]) on MHD Maxwell fluid due to pressure gradient are discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions are converted into dimensionless form. Fractional model for Maxwell fluid is established by Caputo time fractional differential operator. The dimensionless expression for concentration, temperature and velocity are found using Laplace transform. As a result, it is found that fluid properties show dual behavior for small and large time and by increasing volumetric fraction temperature increases and velocity decreases respectively. Further, we compared the Maxwell, Casson and Newtonian fluids and found that Newtonian fluid has greater velocity due to less viscosity. Draw the graphs of temperature and velocity by Mathcad software and discuss the behavior of flow parameters and the effect of fractional parameters.

Three-Dimensional Skyrmions with Arbitrary Topological Number in a Ferromagnetic Spin-1 Bose-Einstein Condensate

We propose a new scheme for creating three-dimensional Skyrmions in a ferromagnetic spin-1 Bose-Einstein condensate by manipulating a multipole magnetic field and a pair of counter-propagating laser beams. The result shows that a three-dimensional Skyrmion with topological number Q = 2 can be created by a sextupole magnetic field and the laser beams. Meanwhile, the vortex ring and knot structure in the Skyrmion are found. The topological number can be calculated analytically in our model, which implies that the method can be extended to create Skyrmions with arbitrary topological number. As the examples, three-dimensional Skyrmions with Q = 3, 4 are also demonstrated and are distinguishable by the density distributions with a specific quantization axis. These topological objects have the potential to be realized in ferromagnetic spin-1 Bose-Einstein condensates experimentally.

Quantum Neimark-Sacker bifurcation

Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on “quantumtorus” and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.

Tunable spinful matter wave valve

We investigate the transport problem that a spinful matter wave is incident on a strong localized spin-orbit-coupled Bose-Einstein condensate in optical lattices, where the localization is admitted by atom interaction only existing at one particular site, and the spin-orbit coupling arouse spatial rotation of the spin texture. We find that tuning the spin orientation of the localized Bose-Einstein condensate can lead to spin-nonreciprocal/spin-reciprocal transport, meaning the transport properties are dependent on/independent of the spin orientation of incident waves. In the former case, we obtain the conditions to achieve transparency, beam-splitting, and blockade of the incident wave with a given spin orientation, and furthermore the ones to perfectly isolate incident waves of different spin orientation, while in the latter, we obtain the condition to maximize the conversion of different spin states. The result may be useful to develop a novel spinful matter wave valve that integrates spin switcher, beam-splitter, isolator, and converter. The method can also be applied to other real systems, e.g., realizing perfect isolation of spin states in magnetism, which is otherwise rather difficult.

Management of matter-wave solitons in Bose-Einstein condensates with time-dependent atomic scattering length in a time-dependent parabolic complex potential

In this paper, we consider a Gross-Pitaevskii (GP) equation with a time-dependent nonlinearity and a spatiotemporal complex linear term which describes the dynamics of matter-wave solitons in Bose-Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms. We establish the integrability conditions under which analytical solutions describing the modulational instability and the propagation of both bright and dark solitary waves on a continuous wave background are obtained. The obtained integrability conditions also appear as the conditions under which the solitary waves of the BECs can be managed by controlling the functional gain or loss parameter. For specific BECs, the dynamics of bright and dark solitons are investigated analytically through the found exact solutions of the GP equation. Our results show that under the integrability conditions, the gain or loss parameter of the GP equation can be used to manage the motion of both bright and dark solitons. We show that for BECs with loss (gain) of atoms, the bright and dark solitons during their propagation have a compression (broadening) in their width. Furthermore, under a safe range of parameters and under the integrability conditions, it is possible to squeeze a bright soliton of BECs with loss of atoms into the assumed peak matter density, which can provide an experimental tool for investigating the range of validity of the 1D GP equation. Our results also reveal that under the conditions of the solitary wave management, neither the injection or the ejection of atoms from the condensate affects the soliton peak during its propagation.

Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap

We consider vortex dynamics in the context of Bose-Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross-Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.

Three-component Gross-Pitaevskii equations in the spin-1 Bose-Einstein condensate: Spin-rotation symmetry, matter-wave solutions, and dynamics

We report new matter-wave solutions of the one-dimensional spin-1 Bose-Einstein condensate system by combining global spin-rotation states and similarity transformation. Dynamical behaviors of non-stationary global spin-rotation states derived from the SU(2) spin-rotation symmetry are discussed, which exhibit temporal periodicity. We derive generalized bright-dark mixed solitons and new rogue wave solutions and reveal the relations between Euler angles in spin-rotation symmetry and parameters in ferromagnetic and polar solitons. In the modulated spin-1 Bose-Einstein condensate system, new solutions are derived and graphically illustrated for different types of modulations. Moreover, numerical simulations are performed to investigate the stability of some obtained solutions for chosen parameters.

Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials

Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields.

Rogue waves in a two-component Manakov system with variable coefficients and an external potential

We construct rogue waves (RWs) in a coupled two-mode system with the self-focusing nonlinearity of the Manakov type (equal SPM and XPM coefficients), spatially modulated coefficients, and a specially designed external potential. The system may be realized in nonlinear optics and Bose-Einstein condensates. By means of a similarity transformation, we establish a connection between solutions of the coupled Manakov system with spatially variable coefficients and the basic Manakov model with constant coefficients. Exact solutions in the form of two-component Peregrine and dromion waves are obtained. The RW dynamics is analyzed for different choices of parameters in the underlying parameter space. Different classes of RW solutions are categorized by means of a naturally introduced control parameter which takes integer values.

Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation

The dynamics of wave packets in the fractional Schrödinger equation is still an open problem. The difficulty stems from the fact that the fractional Laplacian derivative is essentially a nonlocal operator. We investigate analytically and numerically the propagation of optical beams in the fractional Schrödinger equation with a harmonic potential. We find that the propagation of one- and two-dimensional input chirped Gaussian beams is not harmonic. In one dimension, the beam propagates along a zigzag trajectory in real space, which corresponds to a modulated anharmonic oscillation in momentum space. In two dimensions, the input Gaussian beam evolves into a breathing ring structure in both real and momentum spaces, which forms a filamented funnel-like aperiodic structure. The beams remain localized in propagation, but with increasing distance display an increasingly irregular behavior, unless both the linear chirp and the transverse displacement of the incident beam are zero.

Stationary solutions for the nonlinear Schrödinger equation modeling three-dimensional spherical Bose-Einstein condensates in general potentials

Stationary solutions for the cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BECs) confined in three spatial dimensions by general forms of a potential are studied through a perturbation method and also numerically. Note that we study both repulsive and attractive BECs under similar frameworks in order to deduce the effects of the potentials in each case. After outlining the general framework, solutions for a collection of specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular potentials on the behavior of the BECs in these cases, comparing and contrasting the qualitative behavior of the attractive and repulsive BECs for potentials of various strengths and forms. Finally, we consider the nonperturbative where the potential or the amplitude of the solutions is large, obtaining various qualitative results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas-Fermi approximation for the stationary solutions. Naturally, this also occurs in the large mass limit. Through all of these results, we are able to understand the qualitative behavior of spherical three-dimensional BECs in weak, intermediate, or strong confining potentials.

Stationary solutions for the 2+1 nonlinear Schrödinger equation modeling Bose-Einstein condensates in radial potentials

Stationary solutions for the 2+1 cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BEC) in a small potential are obtained via a form of perturbation. In particular, perturbations due to small potentials which either confine or repel the BECs are studied, and under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of radial BEC solutions. Numerical results are also provided for regimes where perturbative results break down (i.e., the large-potential regime). Both repulsive and attractive BECs are considered under this framework. Solutions for many specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular small potentials on the behavior of the BECs.

Stationary solutions for the 1+1 nonlinear Schrödinger equation modeling attractive Bose-Einstein condensates in small potentials

Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation (NLS) modeling attractive Bose-Einstein condensates (BECs) in a small potential are obtained via a form of nonlinear perturbation. The focus here is on perturbations to the bright soliton solutions due to small potentials which either confine or repel the BECs: under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of the bright solitons. Importantly, we do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This is useful, in that it allows us to consider perturbations of bright solitons of arbitrary size. In some cases, exact solutions can be recovered, and these agree with known results from the literature. Several special cases are considered which involve confining potentials of specific relevance to BECs. We make several observations on the influence of the small potentials on the behavior of the perturbed bright solitons. The results demonstrate the difference between perturbed bright solitons in the attractive NLS and those results found in the repulsive NLS for dark solitons, as discussed by Mallory and Van Gorder, [Phys. Rev. E 88 013205 (2013)]. Extension of these results to more spatial dimensions is mentioned.

Abundant soliton solutions of general nonlocal nonlinear Schrödinger system with external field

Periodic and quasi-periodic breather multi-solitons solutions, the dipole-type breather soliton solution, the rogue wave solution, and the fission soliton solution of the general nonlocal Schrödinger equation are derived by using the similarity transformation and manipulating the external potential function. The stability of the exact solitary wave solutions with the white noise perturbation also is investigated numerically.

Transfer of dipolar gas through the discrete localized mode

By considering the discrete nonlinear Schrödinger model with dipole-dipole interactions for dipolar condensate, the existence, the types, the stability, and the dynamics of the localized modes in a nonlinear lattice are discussed. It is found that the contact interaction and the dipole-dipole interactions play important roles in determining the existence, the type, and the stability of the localized modes. Because of the coupled effects of the contact interaction and the dipole-dipole interactions, rich localized modes and their stability nature can exist: when the contact interaction is larger and the dipole-dipole interactions is smaller, a discrete bright breather occurs. In this case, while the on-site interaction can stabilize the discrete breather, the dipole-dipole interactions will destabilize the discrete breather; when both the contact interaction and the dipole-dipole interactions are larger, a discrete kink appears. In this case, both the on-site interaction and the dipole-dipole interactions can stabilize the discrete kink, but the discrete kink is more unstable than the ordinary discrete breather. The predicted results provide a deep insight into the dynamics of blocking, filtering, and transfer of the norm in nonlinear lattices for dipolar condensates.

Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation

A spectral problem, the x-derivative part of which is a simple generalization of the standard Ablowitz-Kaup-Newell-Segur and Kaup-Newell spectral problems, is presented with its associated generalized mixed nonlinear Schrödinger (GMNLS) model. The N-fold Darboux transformation with multi-parameters for the spectral problem is constructed with the help of gauge transformation. According to the Darboux transformation, the solution of the GMNLS model is reduced to solving a linear algebraic system and two first-order ordinary differential equations. As an example of application, we list the modulus formulae of the envelope one- and two-soliton solutions. Note that our model is a generalized one with the inclusion of four coefficients (a, b, c, and d), which involves abundant NLS-type models such as the standard cubic NLS equation, the Gerdjikov-Ivanov equation, the Chen-Lee-Liu equation, the Kaup-Newell equation, and the mixed NLS of Wadati and/or Kundu, among others.

Stationary solutions for the 1+1 nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates in small potentials

Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates (BEC) in a small potential are obtained through a form of nonlinear perturbation. In particular, for sufficiently small potentials, we determine the perturbation theory of stationary solutions, by use of an expansion in Jacobi elliptic functions. This idea was explored before in order to obtain exact solutions [Bronski, Carr, Deconinck, and Kutz, Phys. Rev. Lett. 86, 1402 (2001)], where the potential itself was fixed to be a Jacobi elliptic function, thereby reducing the nonlinear ODE into an algebraic equation, (which could be easily solved). However, in the present paper, we outline the perturbation method for completely general potentials, assuming only that such potentials are locally small. We do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This allows us to consider even poorly behaved potentials, so long as they are bounded locally. We demonstrate the effectiveness of this approach by considering a number of specific potentials: for the simplest potentials, and we recover results from the literature, while for more complicated potentials, our results are new. Dark soliton solutions are constructed explicitly for some cases, and we obtain the known one-soliton tanh-type solution in the simplest setting for the repulsive BEC. Note that we limit our results to the repulsive case; similar results can be obtained for the attractive BEC case.

Phase engineering, modulational instability, and solitons of Gross-Pitaevskii-type equations in 1+1 dimensions

Motivated by recent proposals of "collisionally inhomogeneous" Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we introduce a phase imprint into the macroscopic order parameter governing the dynamics of BECs with spatiotemporal varying scattering length described by a cubic Gross-Pitaevskii (GP) equation and then suitably engineer the imprinted phase to generate the modified GP equation, also called the cubic derivative nonlinear Schrödinger (NLS) equation. This equation describes the dynamics of condensates with two-body (attractive and repulsive) interactions in a time-varying quadratic external potential. We then carry out a theoretical analysis which invokes a lens-type transformation that converts the cubic derivative NLS equation into a modified NLS equation with only explicit temporal dependence. Our analysis suggests a particular interest in a specific time-varying potential with the strength of the magnetic trap ~1/(t+t(*))(2). For a time-varying quadratic external potential of this kind, an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed. We point out the effect of the imprint parameter and the parameter t(*) on the instability growth rate, as well as on the solitary waves of the BECs.

Discrete breather and its stability in a general discrete nonlinear Schrödinger equation with disorder

By considering a general discrete nonlinear Schrödinger model with arbitrary values of nonlinearity power and disorder, the existence and stability of a discrete breather (DB) in a general nonlinear lattice are discussed. It is found that nonlinearity and disorder play important roles in determining the existence and stability of the DB. Nonlinearity (expressed by the interparticle interaction) and disorder can enhance the stability of the DB. Remarkably, we find that the DB is most stable when the nonlinearity power is equal to a critical value. The effects of nonlinearity, nonlinearity power, and disorder on the stability of the DB are strongly coupled.

Wide localized solitons in systems with time- and space-modulated nonlinearities

In this work we apply point canonical transformations to solve some classes of nonautonomous, nonlinear Schrödinger equations, namely, those which possess specific cubic and quintic (time- and space-dependent) nonlinearities. In this way we generalize some procedures recently published which resort to an ansatz to the wave function and recover a time- and space-independent nonlinear equation which can be solved explicitly. The method applied here allows us to find wide localized (in space) soliton solutions to the nonautonomous, nonlinear Schrödinger equation. We also generalize the external potential which traps the system and the terms of the nonlinearities.

Dynamics of analytical three-dimensional solutions in Bose-Einstein condensates with time-dependent gain and potential

Using the F-expansion method we systematically present exact solutions of the three-dimensional nonlinear generalized Gross-Pitaevskii equation, with time-varying gain or loss, in both attractive and expulsive harmonic confinement regimes. This approach allows us to obtain solitons for a large variety of solutions depending on the time-varying potential and the gain or loss profiles. The dynamics of these matter waves, including quasibreathing solitons, double-quasibreathing solitons, and three-quasibreathing solitons, is discussed. The explicit functions that describe the evolution of the amplitude, width, and trajectory of the soliton's wave center are presented exactly. It is demonstrated that an arbitrary additional time-dependent gain function can be added to the model to control the amplitude and width of the soliton and the nonlinearity without affecting the motion of the solitons' wave center. Additionally, a number of exact traveling waves, including the Faraday pattern formation, have been found. The obtained results may raise the possibility of relative experiments and potential applications.

Matter-wave solutions in Bose-Einstein condensates with harmonic and Gaussian potentials

We study exact matter-wave solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the space- and/or time-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation and symbolic analysis, we report several families of exact solutions of the quasi-one-dimensional GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.

Dynamics of kink-dark solitons in Bose-Einstein condensates with both two- and three-body interactions

The matter-wave solutions of Bose-Einstein condensates with three-body interaction are examined through the one-dimensional Gross-Pitaevskii equation. By using a modified lens-type transformation and a further extension of the tanh-function method we obtain the exact analytical solutions which describe the propagation of kink-shaped solitons, anti-kink-shaped solitons, and other families of solitary waves. We realize that the shape of a kink solitary wave depends on both the scattering length and the parameter of atomic exchange with the substrate. The stability of the solitary waves is examined using analytical and numerical methods. Our results can also be applied to nonlinear optics in the presence of cubic-quintic media.

Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates

We report an algorithm of constructing linear and nonlinear potentials in the two-dimensional Gross-Pitaevskii equation subject to given boundary conditions, which allow for exact analytic solutions. The obtained solutions represent superfluid flows in inhomogeneous Bose-Einstein condensates. The method is based on the combination of the similarity reduction of the two-dimensional Gross-Pitaevskii equation to the one-dimensional nonlinear Schrödinger equation, the latter allowing for exact solutions, with the conformal mapping of the given domain, where the flow is considered, to a half space. The stability of the obtained flows is addressed. A number of stable and physically relevant examples are described.

Dynamics of a nonautonomous soliton in a generalized nonlinear Schrödinger equation

We solve a generalized nonautonomous nonlinear Schrödinger equation analytically by performing the Darboux transformation. The precise expressions of the soliton's width, peak, and the trajectory of its wave center are investigated analytically, which symbolize the dynamic behavior of a nonautonomous soliton. These expressions can be conveniently and effectively applied to the management of soliton in many fields.

Control of the symmetry breaking in double-well potentials by the resonant nonlinearity management

We introduce a one-dimensional model of Bose-Einstein condensates (BECs), combining the double-well potential, which is a usual setting for the onset of spontaneous-symmetry-breaking (SSB) effects, and time-periodic modulation of the nonlinearity, which may be implemented by means of the Feshbach-resonance-management (FRM) technique. Both cases of the nonlinearity that is repulsive or attractive on the average are considered. In the former case, the main effect produced by the application of the FRM is spontaneous self-trapping of the condensate in either of the two potential wells in parameter regimes where it would remain untrapped in the absence of the management. In the weakly nonlinear regime, the frequency of intrinsic oscillations in the FRM-induced trapped state is very close to half the FRM frequency, suggesting that the effect is accounted for by a parametric resonance. In the case of the attractive nonlinearity, the FRM-induced effect is the opposite, i.e., enforced detrapping of a state which is self-trapped in its unmanaged form. In the latter case, the frequency of oscillations of the untrapped mode is close to a quarter of the driving frequency, suggesting that a higher-order parametric resonance may account for this effect.

Stability and dynamics of two-soliton molecules

The problem of soliton-soliton force is revisited. From the exact two-soliton solution of a nonautonomous Gross-Pitaevskii equation, we derive a generalized formula for the mutual force between two solitons. The force is given for arbitrary soliton amplitude difference, relative speed, phase, and separation. The latter allows for the investigation of soliton molecule formation, dynamics, and stability. We reveal the role of the time-dependent relative phase between the solitons in binding them in a soliton molecule. We derive its equilibrium bond length, spring constant, frequency, effective mass, and binding energy of the molecule. We investigate the molecule's stability against perturbations such as reflection from surfaces, scattering by barriers, and collisions with other solitons.

Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities

It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrödinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3) space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentrate on power (linear and nonlinear) potentials presenting a number of case examples. Generalizations of the method are also discussed.

Engineering integrable nonautonomous nonlinear Schrödinger equations

We investigate Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painlevé analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painlevé integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.