Gross-Pitaevskii System Research Articles

Dynamics of fractional solitonic profiles to multicomponent Gross–Pitaevskii system

The fractional multicomponent Gross-Pitaevskii system arising in the Bose-Einestein condensate is under consideration. The Gross-Pitaevskii equation plays a significant role in Bose-Einstein condensation and engineering, where it characterizes the dynamics of the condensate wave function. Superfluidity and superconductivity are two characteristics of the low-temperature phenomenon that are linked to the Bose–Einstein condensate, which is generated by a diluted atomic gas. The investigation of multi-component equations has garnered considerable attention because of their capacity to clarify intricate physical phenomena and reveal the dynamic configurations of localized wave solutions. A variety of solutions have been secured in various forms, including bright, dark, singular, and combo solitons, in addition to solutions of hyperbolic, periodic, and exponential functions. For the purpose of ensuring the solutions, recently developed integration tools called the modified Sardar subequation method and enhanced modified extended tanh-expansion method have been implemented. In nonlinear dispersive media, solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The proposed approaches are certainly the most direct, efficient, and valuable method for dealing with multiple nonlinear models that arise in applied physics and mathematics, with the purpose of generating various types of exact solutions. In addition, 3D, 2D, contour, and density plots have been utilized to visually represent the obtained results, facilitating a greater understanding of the physical effects of the derived solutions. The solutions attained are of great importance with regard to their applicability across a wide range of quantum systems.

Spatiotemporal modulated solitons in a quasi-one-dimensional spin-1 Bose–Einstein condensates

In this paper, we investigate the nonautonomous bright/dark solitons in a quasi-one-dimensional spin-1 Bose–Einstein condensates through a three coupled Gross–Pitaevskii (GP) system with space–time-dependent external potential and temporally modulated gain/loss distributions. Based on the Hirota bilinear method, we analytically construct the bright soliton solutions when the coupled GP system exhibits attractive interactions, while we obtain the dark soliton solutions when the coupled GP system exhibits repulsive interactions. The influence of spatiotemporal modulated external potentials, such as the gain/loss distribution Γ(t), on bright/dark soliton dynamics is analyzed in detail via the analytical solutions. By taking different Γ(t), we obtain different types of bright solitons, including periodic, dromion-like and parabolic solitons, and also derive dark solitons on different backgrounds, such as periodic, parabolic and kink backgrounds. We analyze the regulatory effects of different wavenumber ratios on the attraction and squeezing of bound-state solitons. Through the asymptotic analysis, we find that the interactions between two solitons are elastic. In addition, we conduct research on the forward and inverse problems of the above results via the parallel hard-constraint physical informed neural network (phPINN) method. The predicted solitons and potential functions are in good agreement with the exact solitons and potential in the system.

Self-similar evolution of wave turbulence in Gross-Pitaevskii system.

We study the universal nonstationary evolution of wave turbulence (WT) in Bose-Einstein condensates (BECs). Their temporal evolution can exhibit different kinds of self-similar behavior corresponding to a large-time asymptotic of the system or to a finite-time blowup. We identify self-similar regimes in BECs by numerically simulating the forced and unforced Gross-Pitaevskii equation(GPE) and the associated wave kinetic equation(WKE) for the direct and inverse cascades, respectively. In both the GPE and the WKE simulations for the direct cascade, we observe the first-kind self-similarity that is fully determined by energy conservation. For the inverse cascade evolution, we verify the existence of a self-similar evolution of the second kind describing self-accelerating dynamics of the spectrum leading to blowup at the zero mode (condensate) at a finite time. We believe that the universal self-similar spectra found in the present paper are as important and relevant for understanding the BEC turbulence in past and future experiments as the commonly studied stationary Kolmogorov-Zakharov (KZ) spectra.

Multi-solitons in the spin-orbit-coupled spin-1 Bose-Einstein condensates

We find a restrictive integrable reduction for the one-dimensional spin-orbit-coupled spin-1 Bose-Einstein condensate system, which allows us to reduce the three-component Gross-Pitaevskii system into the two-component Manakov system or the single cubic Schrödinger equation. Based on the reduction, we obtain the multi-breathers, higher-order rouge waves and multi-dark soliton solutions for ferromagnetic and anti-ferromagnetic system, respectively. Furthermore, the impacts of the spin-orbit coupling to the multi-soliton dynamics, and the stability of these soliton solutions have also been discussed.

Rigidity of phase transitions for the fractional elliptic Gross-Pitaevskii system

We prove the monotonicity and one-dimensional symmetry of solutions to non-cooperative fractional elliptic systems with uniform limits. In particular, our result holds for phase transitions for the fractional elliptic Gross-Pitaevskii system. The main tool we use is a sliding method adapted to fractional elliptic systems.

Testing wave turbulence theory for the Gross-Pitaevskii system.

We test the predictions of the theory of weak wave turbulence by performing numerical simulations of the Gross-Pitaevskii equation(GPE) and the associated wave-kinetic equation(WKE). We consider an initial state localized in Fourier space, and we confront the solutions of the WKE obtained numerically with GPE data for both the wave-action spectrum and the probability density functions (PDFs) of the Fourier mode intensities. We find that the temporal evolution of the GPE data is accurately predicted by the WKE, with no adjustable parameters, for about two nonlinear kinetic times. Qualitative agreement between the GPE and the WKE persists also for longer times with some quantitative deviations that may be attributed to the combination of a breakdown of the theoretical assumptions underlying the WKE as well as numerical issues. Furthermore, we study how the wave statistics evolves toward Gaussianity in a timescale of the order of the kinetic time. The excellent agreement between direct numerical simulations of the GPE and the WKE provides a solid foundation to the theory of weak wave turbulence.

Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation

IntroductionThe Gross-Pitaevskii equation is a class of the nonlinear Schrödinger equation, whose exact solution, especially soliton solution, is proposed for understanding and studying Bose-Einstein condensate and some nonlinear phenomena occurring in the intersection field of Bose-Einstein condensate with some other fields. It is an important subject to investigate their exact solutions. ObjectivesWe give multi-soliton of a two-dimensional Gross-Pitaevskii system which contains the time-varying trapping potential with a few interactions of multi-soliton. Through analytical and graphical analysis, we obtain one-, two- and three-soliton which are affected by the strength of atomic interaction. The asymptotic expression of two-soliton embodies the properties of solitons. We can give some interactions of solitons of different structures including parabolic soliton, line-soliton and dromion-like structure. MethodsBy constructing an appropriate Hirota bilinear form, the multi-soliton solution of the system is obtained. The soliton elastic interaction is analyzed via asymptotic analysis. ResultsThe results in this paper theoretically provide the analytical bright soliton solution in the two-dimensional Bose-Einstein condensation model and their interesting interaction. To our best knowledge, the discussion and results in this work are new and important in different fields. ConclusionsThe study enriches the existing nonlinear phenomena of the Gross-Pitaevskii model in Bose-Einstein condensation, and prove that the Hirota bilinear method and asymptotic analysis method are powerful and effective techniques in physical sciences and engineering for analyzing nonlinear mathematical-physical equations and their solutions. These provide a valuable basis and reference for the controllability of bright soliton phenomenon in experiments for high-dimensional Bose-Einstein condensation.

Multi-fold binary Darboux transformation and mixed solitons of a three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate

The Bose-Einstein-condensation applications give rise to the superfluidity in the liquid helium and superconductivity in the metals. In this paper, we work on a three-component Gross-Pitaevskii system, which describes the matter waves in an spin-1 spinor Bose-Einstein condensate. We construct a multi-fold binary Darboux transformation with the zero seed solutions to describe the three vertical spin projection of the spin-1 spinor BEC, which is different from all the existing Darboux-type ones for the same system, and derive three types of the exponential-and-rational mixed soliton solutions associated with two conjugate complex eigenvalues. For such mixed solitons, we give their asymptotic expressions, indicating that they consist of the Ieda-Miyakawa-Wadati (IMW)-polar-state or IMW-ferromagnetic solitons but possess the time-dependent velocities. Asymptotically and graphically, interaction mechanisms between the mixed and exponential solitons are classified in six cases, and we exhibit the inelastic and elastic interactions through calculating the modifications of the polarization matrices and phase shifts for the two interacting solitons. We find that both the IMW-polar-state solitons, including the mixed and exponential solitons, can not alter the other soliton’s intensity distribution during the interaction, while the mixed or exponential soliton in the IMW-ferromagnetic state does.

Symmetry and asymmetry of components for elliptic Gross-Pitaevskii system

This article examines nonnegative solutions of the two-component elliptic Gross-Pitaevskii system{ϵ1Δu1=u1(u12+λu22−1)inRn,ϵ2Δu2=u2(u22+λu12−1)inRn, for parameters 1<λ<∞ and ϵ1,ϵ2∈R+. We provide a natural counterpart of the De Giorgi's conjecture, regarding the Allen-Cahn equation, for the above system and we settle it for n≤3. We prove and apply a geometric Poincaré inequality to show that∇u1⋅∇u2=−|∇u1||∇u2|. We study symmetry of components when ϵ1=ϵ2 and asymmetry of components when ϵ1≠ϵ2. In one dimension, we explore explicit profiles of solutions for specific parameters λ,ϵ1,ϵ2. In addition, we provide a gradient estimate, various energy estimates, Liouville theorems, monotonicity formulae and a formalization of surface tensions of the energy for solutions of the above system.Note that when λ=0, the above system becomes decoupled and recovers the Allen-Cahn equation. The De Giorgi conjecture (1978) for the Allen-Cahn equation states that bounded monotone solutions of the Allen-Cahn equation are one-dimensional when n≤8. This conjecture is already settled under certain assumptions. When 0<λ≤1, the system refers to the real-valued Ginzburg-Landau system and nonnegative solutions are trivial.

Gross-Pitaevskii-Poisson model for an ultracold plasma: Density waves and solitons

We introduce 1D and 2D models of a degenerate bosonic gas composed of ions with positive and negative charges (cations and anions). The system may exist in the mean-field condensate state, enabling the competition of the Coulomb coupling, contact repulsion, and kinetic energy of the particles, provided that their effective mass is reduced by means of a lattice potential. The model combines the Gross-Pitaevskii (GP) equations for the two-component wave function of the cations and anions, coupled to the Poisson equation for the electrostatic potential mediating the Coulomb coupling. The contact interaction in the GP system can be derived, in the Thomas-Fermi approximation, from a system of three GP equations, which includes the wave function of heavy neutral atoms. In the system with fully repulsive contact interactions, we construct stable spatially periodic patterns (density waves, DWs). The transition to DWs is identified by analysis of the modulational instability of a uniformly mixed neutral state. The DW pattern, which represents the system's ground state (GS), is predicted by a variational approximation. In 2D, a stable pattern is produced too, with a quasi-1D shape. The 1D system with contact self-attraction in each component produces bright solitons of three types: neutral ones, with fully mixed components; dipoles, with the components separated by the inter-species contact repulsion; and quadrupoles, with a layer of one component sandwiched between side lobes formed by the other. The transition from the neutral solitons to dipoles is accurately modeled analytically. A chart of the GSs of the different types (neutral solitons, dipoles, or quadrupoles) is produced. Different soliton species do not coexist as stable states. Collisions between traveling solitons are elastic for dipole-dipole pairs, while dipole-antidipole ones merge into stable quadrupoles via multiple collisions.

Construction of a solution for the two-component radial Gross-Pitaevskii system with a large coupling parameter

We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a segregated limiting profile, with the difference of its components satisfying a limit scalar PDE. In the case of radial symmetry, under natural non-degeneracy assumptions on a solution of the limit problem, we establish by a perturbation argument its persistence as a solution to the elliptic system.

Riemann–Hilbert approach of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization

ABSTRACT In this paper, we focus on the N-soliton solutions for the coupled nonisospectral Gross–Pitaevskii (GP) system. By using the Riemann–Hilbert approach, analyzing the spectral problem of the Lax pair, a kind of Riemann–Hilbert problem for a system of the coupled nonisospectral GP system is derived, and the N-soliton solutions for the equation under the conditions of irregularity and reflectionless cases also are presented. Moreover, the multi-component coupled nonisospectral GP system is generalized.

Spatially modulated two- and three-component Rabi-coupled Gross–Pitaevskii systems

Vector rogue wave (RW) formation and its dynamics in Rabi-coupled two- and three-species Bose–Einstein condensates with spatially varying dispersion and nonlinearity are studied. For this purpose, we obtain the RW solution of the two- and three-component inhomogeneous Gross–Pitaevskii (GP) systems with Rabi coupling by introducing suitable rotational and similarity transformations. Then, we investigate the effect of inhomogeneity (spatially varying dispersion, trapping potential and nonlinearity) on vector RWs for two different forms of potential strengths, namely periodic (optical lattice) with specific reference to hyperbolic-type potentials and parabolic cylinder potentials. First, we show an interesting oscillating boomeronic behaviour of dark–bright solitons due to Rabi coupling in two-component condensate with constant nonlinearities. Then in the presence of inhomogeneity but in the absence of Rabi coupling we demonstrate the creation of new daughter RWs co-existing with a dark (bright) soliton part in the first (second) component of the two-component GP system. Further, the hyperbolic modulation (sech type) of the parameter along with the Rabi effect leads to the formation of dromion (two-dimensional localized structure) trains even in the (1 + 1) dimensional two-component GP system, which is a striking feature of Rabi coupling with spatial modulation. Next, our study on three-component condensate reveals the fact that the three RWs can be converted into broad-based zero-background RW appearing on top of a bright soliton by introducing spatial modulation only. Further, by including Rabi coupling we observe beating behaviour of solitons with internal oscillations mostly at the wings. Also, we show that by employing parabolic cylinder modulation with model parameter n, one can produce RWs. Our explores that the spatially varying environment leads to the possibility of realization of a two-dimensional nonlinear structure in (1 + 1) dimensional setting.

Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on a finite interval

In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on the finite interval x ∈ [0, L] by extending the Fokas unified approach. The solution of this three-component system can be expressed by means of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex spectral k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly obtained using three spectral functions {s(k), S(k), SL(k)} arising from the initial data and Dirichlet-Neumann boundary conditions at x = 0, L, respectively. The global relation is also presented and used to deduce two distinct but equivalent types of representations [i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel’fand-Levitan-Marchenko (GLM) approach] for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problem on the finite interval can be extended to the ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also give the linearizable boundary conditions for the GLM representations.

On a degree associated with the Gross-Pitaevskii system with a large parameter

In a number of cases we calculate the sum of the degrees of the small positive solutions of the Gross-Pitaevskii system when the interaction is strong.

Non-autonomous bright solitons and their stability in Rabi coupled binary Bose–Einstein condensates

The dynamics of non-autonomous bright matter-wave solitons in Rabi coupled binary Bose–Einstein condensates is explored. By performing a unitary and similarity/lens-type transformation, we reduce the non-autonomous Gross–Pitaevskii (GP) equation into the celebrated Manakov model. Then, we construct the exact bright solitons of the non-autonomous GP system in the presence of time dependent nonlinearities for two specific forms, namely hyperbolic nonlinearities, which are of physical interest. The experimental possibilities of realizing the forms of temporally modulated potentials corresponding to these time dependent nonlinearities and supporting such localized structures are explored. Our study on the propagation of one soliton shows that the amplitude, velocity and shape of the bright soliton are altered by the time-dependent scattering length. We also analyze the non-trivial energy sharing collision of Manakov solitons in the presence of Rabi coupling and aforementioned nonlinearities. We find that breathers arise in two-soliton collisions and the nature of energy sharing collisions is altered from that of Manakov system due to Rabi coupling only. Further in the presence of time-dependent nonlinearities the collision scenario is again altered significantly. Finally, the stability of these localized structures is examined using a recently developed powerful analytic method by Quintero et al (2015 Phys. Rev. E 91, 012905) and it is shown that the non-autonomous bright solitons are indeed stable. The evolution of position and velocity is also studied.

Darboux transformation of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization

In this paper, we extend the one-component Gross–Pitaevskii (GP) equation to the two-component coupled GP system including damping term, linear and parabolic density profiles. The Lax pair with nonisospectral parameter and infinitely-many conservation laws of this coupled GP system are presented. Actually, the Darboux transformation (DT) for this kind of nonautonomous system is essentially different from the autonomous case. Consequently, we construct the DT of the coupled GP equations, besides, nonautonomous multi-solitons, one-breather and the first-order rogue wave are also obtained. Various kinds of one-soliton solution are constructed, which include stationary one-soliton and nonautonomous one-soliton propagating along the negative (positive) direction of x-axis. The interaction of two solitons and two-soliton bound state are demonstrated respectively. We get the nonautonomous one-breather on a curved background and this background is completely controlled by the parameter β. Using a limiting process, the nonautonomous first-order rogue wave can be obtained. Furthermore, some dynamic structures of these analytical solutions are discussed in detail. In addition, the multi-component generalization of GP equations are given, then the corresponding Lax pair and DT are also constructed.

A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstei...

Matter-wave solitons in the counterflow of two immiscible superfluids

We study formation of solitons induced by counterflows of immiscible superfluids. Our setting is based on a quasi-one-dimensional binary Bose-Einstein condensate, composed of two immiscible components with large and small numbers of atoms in them. Assuming that the ``small'' component moves with constant velocity, either by itself, or being dragged by a moving trap, and intrudes into the ``large'' counterpart, the following results are obtained. Depending on the velocity, and on whether the small component moves in the absence or in the presence of the trap, two-component dark-bright solitons, scalar dark solitons, or multiple dark solitons may emerge, the last outcome taking place due to breakdown of the superfluidity. We present two sets of analytical results to describe this phenomenology. In an intermediate velocity regime, where dark-bright solitons form, a reduction of the two-component Gross-Pitaevskii system to an integrable Mel'nikov system is developed, demonstrating that solitary waves of the former are very accurately described by analytically available solitons of the latter. In the high-velocity regime, where the breakdown of the superfluidity induces the formation of dark solitons and multisoliton trains, an effective single-component description, in which a strongly localized wave packet of the ``small'' component acts as an effective potential for the ``large'' one, allows us to estimate the critical velocity beyond which the coherent structures emerge in good agreement with the numerical results.

Integrability and Multi-Soliton Solutions for a Variable-Coefficient Coupled Gross–Pitaevskii System for Atomic MatterWaves

Under investigation in this paper is a variable-coefficient coupled Gross-Pitaevskii (GP) system, which is associated with the studies on atomic matter waves. Through the Painlev´e analysis, we obtain the constraint on the variable coefficients, under which the system is integrable. The bilinear form and multi-soliton solutions are derived with the Hirota bilinear method and symbolic computation. We found that: (i) in the elastic collisions, an external potential can change the propagation of the soliton, and thus the density of the matter wave in the two-species Bose-Einstein condensate (BEC); (ii) in the shape-changing collision, the solitons can exchange energy among different species, leading to the change of soliton amplitudes.We also present the collisions among three solitons of atomic matter waves.