Vanishing Background Research Articles

On the ground state of one-dimensional quantum droplets for large chemical potentials

Abstract In the present work we revisit the problem of the quantum droplet in atomic Bose–Einstein condensates with an eye towards describing its ground state in the large density, so-called Thomas–Fermi (TF) limit. We consider the problem as being separable into 3 distinct regions: an inner one, where the TF approximation is valid, a sharp transition region where the density abruptly drops towards the (vanishing) background value and an outer region which asymptotes to the background value. We analyze the spatial extent of each of these regions, and develop a systematic effective description of the rapid intermediate transition region. Accordingly, we derive a uniformly valid description of the ground state that is found to accurately match our numerical computations. As an additional application of our considerations, we show that this formulation allows for an analytical approximation of excited states such as the (trapped) dark soliton in the large density limit.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

Modulational instability and rogue wave solutions for the mixed focusing–defocusing semi-discrete coherently coupled nonlinear Schrödinger system with 4 × 4 Lax pair

Under consideration is the semi-discrete mixed focusing–defocusing coherently coupled nonlinear Schrödinger system with the fourth-order auxiliary linear problem, which may describe some dynamic behaviors of light pulses. Firstly, the modulational instability is studied to analyze the possible generation reason of diverse localized waves from plane wave solution. Secondly, based on the known 4 × 4 Lax pair, the discrete generalized (m,N-m)-fold Darboux transformation is constructed and extended to solve this discrete system. Finally, as an application of the resulting Darboux transformation, some exact solutions with position control parameters are derived and shown graphically, in particular, we obtain some novel rogue wave and periodic wave solutions with only one peak and no valleys on vanishing background, which are different from the usual fundamental rogue wave with one peak and two valleys. These results might be useful for understanding propagation of light pulses.

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.

Darboux transformation and general soliton solutions for the reverse space–time nonlocal short pulse equation

We construct the Darboux transformation for the reverse space–time (RST) nonlocal short pulse equation by considering a nonlocal symmetry reduction of the Ablowitz–Kaup–Newell–Segur spectral problem and a covariant hodograph transformation. Some essential differences between the RST nonlocal models and the usual (local) ones are demonstrated by means of a series of explicit solutions and the solving process. The multi-bright soliton solution on vanishing background, as well as the multi-dark soliton, multi-breather and higher-order rogue wave solutions corresponding to nonvanishing background of the RST nonlocal defocusing short pulse equation are derived through the Darboux transformation. The classification and dynamics of these explicit solutions with loop-, cuspon- and smooth-type are presented. The asymptotic analysis is rigorously performed for the two-bright soliton, two-dark soliton and two-breather solutions. Some novel wave patterns such as the double-loop and mixed loop-cuspon rogue waves which are different from their regular counterparts are shown. Other types of singular solutions along certain space–time lines are also obtained.

Schwarzschild quasi-normal modes of non-minimally coupled vector fields

We study perturbations of massive and massless vector fields on a Schwarzschild black-hole background, including a non-minimal coupling between the vector field and the curvature. The coupling is given by the Horndeski vector-tensor operator, which we show to be unique, also when the field is massive, provided that the vector has a vanishing background value.We determine the quasi-normal mode spectrum of the vector field, focusing on the fundamental mode of monopolar and dipolar perturbations of both even and odd parity, as a function of the mass of the field and the coupling constant controlling the non-minimal interaction. In the massless case, we also provide results for the first two overtones, showing in particular that the isospectrality between even and odd modes is broken by the non-minimal gravitational coupling.We also consider solutions to the mode equations corresponding to quasi-bound states and static configurations. Our results for quasi-bound states provide strong evidence for the stability of the spectrum, indicating the impossibility of a vectorization mechanism within our set-up. For static solutions, we analytically and numerically derive results for the electromagnetic susceptibilities (the spin-1 analogs of the tidal Love numbers), which we show to be non-zero in the presence of the non-minimal coupling.

Dynamical behaviors of optical soliton for integrable three-component coupled nonlinear Schrödinger equation

In this paper, a novel form of optical soliton solution for the integrable three-component coupled nonlinear Schrödinger equation on the vanishing background is investigated. Using algebraic iteration and generalized Darboux transformation, optical soliton solutions of single- to four-solitons are constructed. On the basis of above solutions, new elastic and inelastic collisions of nonlinear wave are displayed, especially interactions of four-solitons. It is shown that free parameters play a crucial role in dynamical properties of solitons. With the change of free parameters, some novel three-dimensional diagrams of interactions for three- and four-solitons are illustrated by using numerical simulation, respectively. In addition, new bound states of four-solitons are also studied for the first time.

Compensation process and generation of chirped femtosecond solitons and double-kink solitons in Bose–Einstein condensates with time-dependent atomic scattering length in a time-varying complex potential

We consider the one-dimensional (1D) cubic-quintic Gross–Pitaevskii (GP) equation, which governs the dynamics of Bose–Einstein condensate matter waves with time-varying scattering length and loss/gain of atoms in a harmonic trapping potential. We derive the integrability conditions and the compensation condition for the 1D GP equation and obtain, with the help of a cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift, exact analytical solitonlike solutions with the corresponding frequency chirp which describe the dynamics of femtosecond solitons and double-kink solitons propagating on a vanishing background. Our investigation shows that under the compensation condition, the matter wave solitons maintain a constant amplitude, the amplitude of the frequency chirp depends on the scattering length, while the motion of both the matter wave solitons and the corresponding chirp depend on the external trapping potential. More interesting, the frequency chirps are localized and their feature depends on the sign of the self-steepening parameter. Our study also shows that our exact solutions can be used to describe the compression of matter wave solitons when the absolute value of the s-wave scattering length increases with time.

Instability of compact stars with a nonminimal scalar-derivative coupling

For a theory in which a scalar field ϕ has a nonminimal derivative coupling to the Einstein tensor Gμ ν of the form ϕ Gμ ν∇μ∇ν ϕ, it is known that there exists a branch of static and spherically-symmetric relativistic stars endowed with a scalar hair in their interiors. We study the stability of such hairy solutions with a radial field dependence ϕ(r) against odd- and even-parity perturbations. We show that, for the star compactness \U0001d49e smaller than 1/3, they are prone to Laplacian instabilities of the even-parity perturbation associated with the scalar-field propagation along an angular direction. Even for \U0001d49e>1/3, the hairy star solutions are subject to ghost instabilities. We also find that even the other branch with a vanishing background field derivative is unstable for a positive perfect-fluid pressure, due to nonstandard propagation of the field perturbation δ ϕ inside the star. Thus, there are no stable star configurations in derivative coupling theory without a standard kinetic term, including both relativistic and nonrelativistic compact objects.

Dynamics of multi-soliton and breather solutions for a new semi-discrete coupled system related to coupled NLS and coupled complex mKdV equations

In this paper, a new semi-discrete coupled system which was firstly proposed by Bronsard and Pelinovsky is under investigation. Based on its known Lax pair, the infinitely-many conservation laws and discrete N-fold DT for this system are constructed. As applications, bell-shaped multi-soliton and breather solutions in terms of determinants for this system are firstly derived by means of the discrete N-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically: (1) Propagation characteristics of one-, two-, three- and four-soliton solutions are discussed from vanishing background. (2) Propagation characteristics of one- and two-breather solutions are analyzed from the plane wave background. The details of the dynamical evolutions for such soliton and breather solutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of these solitons with or without a noise in a relatively short period of time, while the evolutions exhibit obviously larger oscillations and strong instability with the increase in time. These results may be useful for understanding the propagation of orthogonally polarized optical waves in an isotropic medium and circularly polarized few-cycle pulses in Kerr media described by the coupled NLS and coupled complex mKdV equations, respectively.

N-fold Darboux transformation and discrete soliton solutions for the discrete Hirota equation

In this paper, a discrete Hirota equation is analytically investigated. The N-fold Darboux transformation (DT) is constructed based on the Lax pair for the equation. In addition, the discrete N-soliton solutions under the vanishing background are derived. Especially, the dynamic features of one-soliton and two-soliton solutions are displayed through figures.

Properties of a Rare-Earth-Ion-Doped Waveguide at Sub-Kelvin Temperatures for Quantum Signal Processing

We characterize the 795nm ^{3}H_{6} to ^{3}H_{4} transition of Tm^{3+} in a Ti^{4+}:LiNbO_{3} waveguide at temperatures as low as 800mK. Coherence and hyperfine population lifetimes-up to 117 μs and 2.5h, respectively-exceed those at 3K at least tenfold, and are equivalent to those observed in a bulk Tm^{3+}:LiNbO_{3} crystal under similar conditions. We also find a transition dipole moment that is equivalent to that of the bulk. Finally, we prepare a 0.5GHz-bandwidth atomic frequency comb of finesse >2 on a vanishing background. These results demonstrate the suitability of rare-earth-ion-doped waveguides created using industry-standard Ti indiffusion in LiNbO_{3} for on-chip quantum applications.

Yang-Mills correlators across the deconfinement phase transition

We compute the finite temperature ghost and gluon propagators of Yang-Mills theory in the Landau-DeWitt gauge. The background field that enters the definition of the latter is intimately related with the (gauge-invariant) Polyakov loop and serves as an equivalent order parameter for the deconfinement transition. We use an effective gauge-fixed description where the nonperturbative infrared dynamics of the theory is parametrized by a gluon mass which, as argued elsewhere, may originate from the Gribov ambiguity. In this scheme, one can perform consistent perturbative calculations down to infrared momenta, which have been shown to correctly describe the phase diagram of Yang-Mills theories in four dimensions as well as the zero-temperature correlators computed in lattice simulations. In this article, we provide the one-loop expressions of the finite temperature Landau-DeWitt ghost and gluon propagators for a large class of gauge groups and present explicit results for the SU(2) case. These are substantially different from those previously obtained in the Landau gauge, which corresponds to a vanishing background field. The nonanalyticity of the order parameter across the transition is directly imprinted onto the propagators in the various color modes. In the SU(2) case, this leads, for instance, to a cusp in the electric and magnetic gluon susceptibilities as well as similar signatures in the ghost sector. We mention the possibility that such distinctive features of the transition could be measured in lattice simulations in the background field gauge studied here.

On the Asymptotic Stability of $${N}$$ N -Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

We consider the Cauchy problem for the defocusing nonlinear Schrodinger (NLS) equation for finite density type initial data. Using the \({\overline{\partial}}\) generalization of the nonlinear steepest descent method of Deift and Zhou, we derive the leading order approximation to the solution of NLS for large times in the solitonic region of space–time, \({|x| < 2 t}\), and we provide bounds for the error which decay as \({ t \rightarrow \infty}\) for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives. Using properties of the scattering map of NLS we derive, as a corollary, an asymptotic stability result for initial data that are sufficiently close to the N-dark soliton solutions of NLS.

Interplay of Boltzmann Equation and Continuity Equation for Accelerated Electrons in Solar Flares

During solar flares a large number of electrons are accelerated within the plasma present in the solar atmosphere. Accurate measurements of the motion of these electrons start becoming available from the analysis of hard X-ray imaging-spectroscopy observations. In this paper, we discuss the linearized perturbations of the Boltzmann kinetic equation describing an ensemble of electrons accelerated by the energy release occurring during solar flares. Either in the limit of high energy or at vanishing background temperature such an equation reduces to a continuity equation equipped with an extra force of stochastic nature. This stochastic force is actually described by the well known energy loss rate due to Coulomb collision with ambient particles in the limit of vanishing temperature, but, in order to match the collision kernel in the linearized Boltzmann equation it needs to be treated in a very specific manner. We furthermore show that when the background temperature is not negligible a diffusive term appears in the asymptotic expansion of the collision kernel and, in this regime, the approximated dynamics furnished by the continuity equation is not anymore reliable. In the second part of the paper the derived continuity equation is solved with some hyperbolic techniques, and the obtained solution is written in a form suitable to be compared with data gathered by hard X-ray imaging-spectroscopy telescopes. Finally, a first validation of the model with NASA Reuven Ramaty High Energy Solar Spectroscopic Imager measurements is provided.

Wave instabilities in the presence of non vanishing background in nonlinear Schrödinger systems.

We investigate wave collapse ruled by the generalized nonlinear Schrödinger (NLS) equation in 1+1 dimensions, for localized excitations with non-zero background, establishing through virial identities a new criterion for blow-up. When collapse is arrested, a semiclassical approach allows us to show that the system can favor the formation of dispersive shock waves. The general findings are illustrated with a model of interest to both classical and quantum physics (cubic-quintic NLS equation), demonstrating a radically novel scenario of instability, where solitons identify a marginal condition between blow-up and occurrence of shock waves, triggered by arbitrarily small mass perturbations of different sign.

Real-time and real-space spin and energy dynamics in one-dimensional spin-12systems induced by local quantum quenches at finite temperatures

We study the spin and energy dynamics in one-dimensional spin-$\frac{1}{2}$ systems induced by local quantum quenches at finite temperatures using a time-dependent density matrix renormalization group method. System sizes are chosen large enough to ensure that the time-dependent data for the accessible time scales represent the behavior in the thermodynamic limit. As a main result, we observe a ballistic spreading of perturbations of the energy density in the integrable spin-$\frac{1}{2}$ XXZ chain for all temperatures and exchange anisotropies, related to the divergent thermal conductivity in this model and the exact conservation of the energy current. In contrast, the spin dynamics is ballistic in the massless phase, but shows a diffusive behavior at high temperatures in the easy-axis phase in the case of a vanishing background spin density. We extract a quantitative estimate for the spin-diffusion constant from the time dependence of the spatial variance of the spin density, which agrees well with values obtained from current-current correlation functions using an Einstein relation. Interestingly, the diffusion constant approaches a constant value deep in the easy-axis regime. As an example for nonintegrable models, we consider two-leg ladders, for which we observe indications of diffusive energy and spin dynamics. The relevance of our results for recent experiments with quantum magnets and bosons in optical lattices is discussed.

A generalized mathematical model of geostrophic adjustment and frontogenesis: uniform potential vorticity

AbstractFronts, or regions with strong horizontal density gradients, are ubiquitous and dynamically important features in the ocean and atmosphere. In the atmosphere, fronts are associated with some of the most severe weather events, while in the ocean, fronts are associated with enhanced turbulence, water mass transformation and biological activity. Here, we examine the dynamics involved in the formation of fronts, or frontogenesis, in detail using a generalized mathematical framework. This extends previous work which has generally revolved around two limiting cases: fronts generated through forcing due to a convergent large-scale flow, and fronts generated spontaneously during the geostrophic adjustment of an initially unbalanced flow. Here, we introduce a new generalized momentum coordinate to simultaneously describe forced and spontaneous frontogenesis. The nonlinear, inviscid, Boussinesq, hydrostatic governing equations for uniform PV flow are solved for arbitrary Rossby and Froude number. The solution is then examined in three distinct cases. Firstly, for a zero potential vorticity (PV) flow bounded by rigid lids, a general solution is derived for the transient response of the fluid to an arbitrary initial mass imbalance and deformation field. The deformation frontogenesis solution of Hoskins &amp; Bretherton (J. Atmos. Sci., vol. 29, 1972, pp. 11–37) and the mass imbalance solution of Blumen (J. Phys. Oceanogr., vol. 30, 2000, pp. 31–39) emerge as two limits of this general solution. Secondly, the problem of geostrophic adjustment of an initial mass imbalance (no deformation field) is considered for uniform PV flow bounded by rigid lids. The general solution is derived, composed of an adjusted state and a transient component describing the propagation of inertia–gravity waves. The criteria for the occurrence of a frontal discontinuity is determined in terms of the Rossby and Froude numbers. The uniform PV solution reduces identically to the zero PV solution of Blumen in the limit of vanishing background stratification. Thirdly, we examine the more general case of uniform PV flow with a deformation field and either balanced or unbalanced initial conditions. In this case the solution is composed of a time-varying mean state – matching the Hoskins &amp; Bretherton solution in the limit of small strain – and an inertia gravity wave field, the dynamics of which are examined in detail. Our analysis provides a unifying framework capable of describing frontal formation and geostrophic adjustment in a wide variety of settings.

Holographic anyonic superfluidity

Starting with a holographic construction for a fractional quantum Hall state based on the D3-D7' system, we explore alternative quantization conditions for the bulk gauge fields. This gives a description of a quantum Hall state with various filling fractions. For a particular alternative quantization of the bulk gauge fields, we obtain a holographic anyon fluid in a vanishing background magnetic field. We show that this system is a superfluid, exhibiting the relevant gapless excitation.

Spatial solitons supported by localized gain in nonlinear optical waveguides

We introduce a modification of the complex Ginzburg-Landau (CGL) equation with background linear loss and locally applied gain. The equation appertains to laser cavities based on planar waveguides, and also to the description of thermal convection in binary fluids. With the gain localization accounted for by the delta-function, a solution for pinned solitons is found in an analytical form, with one relation imposed on parameters of the model. The exponentially localized solution becomes weakly localized in the limit case of vanishing background loss. Numerical solutions, with the delta-function replaced by a finite-width approximation, demonstrate stability of the pinned solitons and their existence in the general case, when the analytical solution is not available. If the gain-localization region and the size of the soliton are comparable, the static soliton is replaced by a stable breather.