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.     

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.     

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.     

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.     

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

By symbolic computation and a direct method, this paperpresents some exact analytical solutions of the one-dimensionalgeneralized inhomogeneous higher-order nonlinear Schr?dingerequation with variable coefficients, which include bright solitons,dark solitons, combined solitary wave solutions, dromions,dispersion-managed solitons, etc. The abundant structure of thesesolutions are shown by some interesting figures with computersimulation.     

Dynamics of Bose-Einstein condensates near Feshbach resonance in external potential

We review our recent theoretical advances in the dynamics of Bose-Einstein condensates with tunable interactions using Feshbach resonance and external potential. A set of analytic and numerical methods for Gross-Pitaevskii equations are developed to study the nonlinear dynamics of Bose-Einstein condensates. Analytically, we present the integrable conditions for the Gross-Pitaevskii equations with tunable interactions and external potential, and obtain a family of exact analytical solutions for one- and two-component Bose-Einstein condensates in one and two-dimensional cases. Then we apply these models to investigate the dynamics of solitons and collisions between two solitons. Numerically, the stability of the analytic exact solutions are checked and the phenomena, such as the dynamics and modulation of the ring dark soliton and vector-soliton, soliton conversion via Feshbach resonance, quantized soliton and vortex in quasi-two-dimensional are also investigated. Both the exact and numerical solutions show that the dynamics of Bose-Einstein condensates can be effectively controlled by the Feshbach resonance and external potential, which offer a good opportunity for manipulation of atomic matter waves and nonlinear excitations in Bose-Einstein condensates.     

THE NONLINEAR SCHRÖDINGER EQUATION FOR THE DELTA-COMB POTENTIAL: QUASI-CLASSICAL CHAOS AND BIFURCATIONS OF PERIODIC STATIONARY SOLUTIONS

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.     

Exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The Bäcklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.     

Etude de la restauration de l'aluminium irradié aux electrons (Etape III)

The nature of the electron bombarded aluminium electrical recovery process has been studied by combining quenches with irradiations. Measured Frenkel pair formation rates per incident electrons as well as annealing kinetics emphasize the role of impurity interstitial trapping.     

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.     

A Generalized Method and Exact Solutions in Bose-Einstein Condensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schr(o)dinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function's soliton solutions, but also Jacobi elliptic function solutions and Weierstrass elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient.     

Exact solutions to three-dimensional time-dependent Schrödinger equation

With a view to obtain exact analytic solutions to the time-dependent Schrödinger equation for a few potentials of physical interest in three dimensions, transformation-group method is used. Interestingly, the integrals of motion in the new coordinates turn out to be the desired invariants of the systems.     

Exact analytical solutions for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients

Analytical solutions are reported for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients using the similarity transformation and tri-function method, which involve three free functions of spaces to generate abundant wave structures. Three types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.     

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 ($c_2>0$).