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.     

Azimuthal modulational instability of vortices in the nonlinear Schrödinger equation

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady-state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize such solutions.     

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.     

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.     

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.     

Integrability of the Gross-Pitaevskii equation with Feshbach resonance management

In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schrödinger equation. By this transformation, each exact solution of the standard nonlinear Schrödinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.     

Matter-wave vortices and solitons in anisotropic optical lattices

Using numerical methods, we construct families of vortical, quadrupole, and fundamental solitons in a two-dimensional (2D) nonlinear-Schrödinger/Gross-Pitaevskii equation which models Bose-Einstein condensates (BECs) or photonic crystals. The equation includes the attractive or repulsive cubic nonlinearity and an anisotropic periodic potential. Two types of anisotropy are considered, accounted for by the difference in the strengths of the 1D sublattices, or by a difference in their periods. The limit case of the quasi-1D optical lattice (OL), when one sublattice is missing, is included too. By means of systematic simulations, we identify stability limits for two species of vortex solitons and quadrupoles, of the rhombus and square types. In the attraction model, rhombic vortices and quadrupoles remain stable up to the limit case of the quasi-1D lattice. In the same model, finite stability limits are found for vortices and quadrupoles of the square type, in terms of the anisotropy parameter. In the repulsion model, rhombic vortices and quadrupoles are stable in large parts of the first finite bandgap (FBG). Another species of partly stable anisotropic states is found in the second FBG, subfundamental dipoles, each squeezed into a single cell of the OL. Square-shaped quadrupoles are completely unstable in the repulsion model, while vortices of the same type are stable only in weakly anisotropic OL potentials.     

Exact spatial soliton solution for nonlinear Schrödinger equation with a type of transverse nonperiodic modulation

In this paper, we analyze (2 + 1)D nonlinear Schrödinger (NLS) equation based on a type of nonperiodic modulation of linear refractive index in the transverse direction. We obtain an exact solution in explicit form for the (2 + 1)D nonlinear Schrödinger (NLS) equation with the nonperiodic modulation. Finally, the stability of the solution is discussed numerically, and the results reveal that the solution is stable to the finite initial perturbations.     

Analytical study of the splitting process of a multiply-quantized vortex in a Bose–Einstein condensate and collaboration of the zero and complex modes

We study the dynamics of a trapped Bose–Einstein condensate with a multiply-quantized vortex, and investigate the roles of the fluctuations in the dynamical evolution of the system. Using the perturbation theory of the external potential, and assuming the situation of the small coupling constant of self-interaction, we analytically solve the time-dependent Gross–Pitaevskii equation. We introduce the zero mode and its adjoint mode of the Bogoliubov–de Gennes equations. Those modes are known to be essential for the completeness condition. We confirm how the complex eigenvalues induce the vortex splitting. It is shown that the physical role of the adjoint zero mode is to ensure the conservation of the total condensate number. The contribution of the adjoint mode is exponentially enhanced in synchronism with the exponential growth of the complex mode, and is essential in the vortex splitting.     

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.     

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.     

Collapse of triaxial bright solitons in atomic Bose-Einstein condensates

We study triaxial bright solitons made of attractive Bose-condensed atoms characterized by the absence of confinement in the longitudinal axial direction but trapped by an anisotropic harmonic potential in the transverse plane. By numerically solving the three-dimensional Gross-Pitaevskii equation we investigate the effect of the transverse trap anisotropy on the critical interaction strength above which there is the collapse of the condensate. The comparison with previous predictions [A. Gammal, L. Tomio, T. Frederico, Phys. Rev. A 66 (2002) 043619] shows significant differences for large anisotropies.     

Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications

In this paper we study blow-up phenomena in general coupled nonlinear Schrödinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose-Einstein condensates and to degenerate boson-fermion mixtures.     

Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrödinger equation

We investigate the properties of modulational instability and discrete breathers in the cubic-quintic discrete nonlinear Schrödinger equation. We analyze the regions of modulational instabilities of nonlinear plane waves. Using the Page approach [J.B. Page, Phys. Rev. B 41 (1990) 7835], we derive the conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The application to the existence of localized modes in the Bose-Einstein condensate (BEC) with three-body interactions in an optical lattice is discussed. The numerical simulations agree with the analytical predictions.     

Modulating the amplitude of dark soliton by scattering-length management in Bose-Einstein condensates

We present a family of soliton solutions of the quasi-one-dimensional Bose-Einstein condensates with time-dependent scattering length, by developing multiple-scale method combined with truncated Painlevé expansion. Then, by numerical calculating the solutions, it is shown that there exhibit two types of dark solitons—black soliton (the zero minimum amplitude at its center) and gray soliton (the minimum density does not drop to zero) in a repulsive condensate. Furthermore, we propose experimental protocols to realize the exchange between black and gray solitons by varying the scattering length via the Feshbach resonance in currently experimental conditions.     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

Chirped and chirpfree soliton solutions of generalized nonlinear Schrödinger equation with distributed coefficients

Using the F-expansion method, we systematically present exact solutions of the generalized nonlinear nonlinear Schrödinger equation with varying intermodal dispersion and nonlinear gain or loss. This approach allows us to obtain large variety of solutions in terms of Jacobi-elliptical and Weierstrass-elliptical functions. The chirped and unchirped spatiotemporal soliton solutions and trigonometric-function solutions have been also obtained as limiting cases. The dynamics of these spatiotemporal soliton is discussed in context of optical fiber communication. To visualize the propagation characteristics of chirp and unchirped dark-bright soliton solutions, few numerical simulations are given. It is found that wave profile of solitons depend on the group velocity dispersion and the gain or loss functions.     

Bifurcation structures and dominant modes near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation

We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrödinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose-Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria.     

Solving nonlinear Schrödinger equation based on discrete split-step multi-wavelet method

In this paper, the discrete split-step multi-wavelet method (DSSMWM) is used to solve nonlinear Schrödinger equation. When the relative amplitude error is below10⁻³magnitude, relative error of amplitude evolution, relative error of pulse broadening ratio, relative phase error, and computing time is respectively achieved. Because multi-wavelet is extraordinary effective for data compression, it only needs to deal with very little data. It can be seen that although the relative amplitude error, relative error of amplitude evolution, relative error of pulse broadening rate and relative phase error changes little, but the computing time are greatly reduced.