Painlevé integrability of a generalized fifth-order KdV equation with variable coefficients: Exact solutions and their interactions

By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painlevé test for integrability only for three distinct cases. Moreover, the multisoliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.     

Spatiotemporal self-similar solutions for the nonautonomous （3＋1）-dimensional cubic-quintic Gross-Pitaevskii equation

With the help of similarity transformation,we obtain analytical spatiotemporal self-similar solutions of the nonautonomous(3+1)-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction,nonlinearity,harmonic potential and gain or loss when two constraints are satisfied.These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction,nonlinearity and the gain/loss.Based on these analytical results,we investigate the dynamic behaviours in a periodic distributed amplification system.     

Chaos in a Bose-Einstein condensate

It is demonstrated that Smale-horseshoe chaos exists in the time evolution of the one-dimensional Bose-Einstein condensate driven by time-periodic harmonic or inverted-harmonic potential.A formally exact solution of the time-dependent Gross-Pitaevskii equation is constructed,which describes the matter shock waves with chaotic or periodic amplitudes and phases.     

Some exact solutions to the inhomogeneous higher-order nonlinear Schrdinger equation by a direct method

By symbolic computation and a direct method, this paper presents some exact analytical solutions of the one-dimensional generalized inhomogeneous higher-order nonlinear Schr?dinger equation with variable coefficients, which include bright solitons, dark solitons, combined solitary wave solutions, dromions, dispersion-managed solitons, etc. The abundant structure of these solutions are shown by some interesting figures with computer simulation.     

Soliton Solutions,Bcklund Transformations and Lax Pair for a(3 + 1)-Dimensional Variable-Coefficient Kadomtsev–Petviashvili Equation in Fluids

Under investigation in this paper is a(3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomials,symbolic computation and auxiliary independent variable, the bilinear forms, soliton solutions, B¨acklund transformations and Lax pair are obtained. Variable coefficients of the equation can affect the solitonic structure, when they are specially chosen, while curved and linear solitons are illustrated. Elastic collisions between/among two and three solitons are discussed, through which the solitons keep their original shapes invariant except for some phase shifts.     

Three-dimensional Bose–Einstein condensate vortex soliton sunder optical lattice and harmonic confinements

We predict three-dimensional vortex solitons in a Bose-Einstein condensate under a complex potential which is the combination of a two-dimensional parabolic trap along the transverse radial direction and a one-dimensional optical-lattice potential along the z axis direction. The vortex solitons are built in the form of layer-chain structure made up of several fundamental vortices along the optical-lattice direction, which were not reported before in the three-dimensional Bose-Einstein condensate. By using the combination of the energy density functional method with the direct numerical simulation, we find three-dimensional vortex solitons with topological charge χ=1, χ=2, and χ=3. Moreover, the macroscopic quantum tunneling and the chirp phenomena of the vortex solitons are shown in the evolution. Thereinto, the occurrence of the macroscopic quantum tunneling provides a possibility for the realization of the quantum tunneling in experiment. Specifically, we manipulate the vortex solitons along the optical lattice direction successfully. The stability limits for dragging the vortex solitons from an initial fixed position to a prescribed location are further pursued.     

Topological aspect of vortex lines in two-dimensional Gross—Pitaevskii theory

Using the -mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the δ function indicates that the vortices can only be generated from the zero points of Φ and are quantized in terms of the Hopf indices and Brouwer degrees. The -mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.     

The dynamics of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates

This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates.It gives three kinds of stationary solutions to this model and develops a general method of constructing nonstationary solutions.It obtains the unique features about general evolution and soliton evolution of nonstationary solutions in this model.     

Exact solutions for nonlinear Schr?dinger equation in phase space: applications to Bose－Einstein condensate

The stationary-state nonlinear Schr?dinger equation, which models the dilute-gas Bose－Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The eigenfunctions in position and momentum spaces are obtained through the ‘Fourier-like' projection transformation from the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.     

Some exact solutions to the inhomogeneous higher-order nonlinear Schrodinger equation by a direct method

By symbolic computation and a direct method, this paper presents some exact analytical solutions of the one-dimensional generalized inhomogeneous higher-order nonlinear Schrodinger equation with variable coefficients, which include bright solitons, dark solitons, combined solitary wave solutions, dromions, dispersion-managed solitons, etc. The abundant structure of these solutions are shown by some interesting figures with computer simulation.     

有弱偏置磁场及含时激光场中的非线性Gross-Pitaevskii方程新的精确解

利用映射方法和一个适当的变换，得到大量的有弱偏置磁场及含时激光场中的非线性Gross-Pitaevskii方程的新解，这些解包括椭圆函数解，椭圆函数叠加解，三角函数解，亮孤子解，暗孤子解和类孤子解。     

Classifying the ground-state phases of spin–orbit coupled spin-2 Bose–Einstein condensate in momentum space

The ground-state phases of two-dimensional spin-2 Bose–Einstein condensate with Rashba spin–orbit coupling are studied. For the equal strengths of the density-density interaction and the spin-exchange interaction, we classify the ground-state phases into four types of stable phases with spin–orbit coupling and spin singlet-pairing interaction in momentum space, i.e., the ring phase, the stripe phase, the triangular phase and the square phase. With increasing the spin–orbit coupling strength, the system undergoes a sequence phase transitions from the ring phase to the stripe phase, and to the square phase for the attractive spin singlet-pairing interaction ($c_2<0$), and the system undergoes a sequence phase transitions from the ring phase to the stripe phase, to the triangular phase, and to the square phase for the repulsive spin singlet-pairing interaction ($c_2>0$).     

Exact Analytical Solutions in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length

In the paper, the generalized Riccati equation rational expansion method is presented. Making use of the method and symbolic computation, we present three families of exact analytical solutions of Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Then the dynamics of two anlytical solutions are demonstrated by computer simulations under some selectable parameters including the Feshbach-managed nonlinear coefficient and the hyperbolic secant function coefficient.     

Exact solutions of the nonlocal Gerdjikov-Ivanov equation

The nonlocal nonlinear Gerdjikov-Ivanov (GI) equation is one of the most important integrable equations, which can be reduced from the third generic deformation of the derivative nonlinear Schrödinger equation. The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation. As applications, we obtain the bright-dark soliton, breather, rogue wave, kink, W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation. These solutions show rich wave structures for selections of different parameters. In all these instances we practically show that these solutions have different properties than the ones for local case.     

CONSTRUCTION OF MODULATED AMPLITUDE WAVES VIA AVERAGING IN COLLISIONALLY INHOMOGENEOUS BOSE–EINSTEIN CONDENSATES

We apply the averaging method to analyze spatio-temportal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasi-one-dimensional collisionally inhomogeneous Bose–Einstein condensates with the scattering length varying periodically in space and crossing zero. Infinitely many modulated amplitude waves with nontrivial phases are shown.     

Padé approximations of quantized-vortex solutions of the Gross–Pitaevskii equation

Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padéapproximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that(1) they approximate the entire solutions quite well from the core to infinity;(2) higher-order Padé approximate solutions have higher accuracy;(3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with ~(87)Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.     

Formations of n-order two-soliton bound states in Bose–Einstein condensates with spatiotemporally modulated nonlinearities

The formations of n-order two-soliton bound states (TSBSs) in the Bose–Einstein condensates with spatiotemporally modulated nonlinearities are studied. Exact analytical expressions of the n-order TSBSs are derived by means of the similarity transformations. Further, the numerical simulations are carried out, consistent with the analytical results very well. The stability analysis shows that the solutions can be stable. Our results indicate that the attractive spatiotemporal inhomogeneous nonlinearities can support n-order TSBSs, which has the potential applications to the generation of matter-wave bright solitons in harmonic traps.