Quintic Terms Research Articles

One-dimensional cubic-quintic Gross-Pitaevskii equation for Bose-Einstein condensates in a trap potential

By means of new general variational method we report a direct solution for the quintic self-focusing nonlinearity and cubic-quintic 1D Gross Pitaeskii equation (GPE) in a harmonic confined potential. We explore the influence of the 3D transversal motion generating a quintic nonlinear term on the ideal 1D pure cigar-like shape model for the attractive and repulsive atom-atom interaction in Bose Einstein condensates (BEC). Also, we offer a closed analytical expression for the evaluation of the error produced when solely the cubic nonlinear GPE is considered for the description of 1D BEC.

Rogue waves of the nonlinear Schrödinger equation with even symmetric perturbations

We show that a rogue wave solution of the nonlinear Schrödinger equation (NLSE) can survive even-parity perturbations of the equation, such as the addition of a quintic term and fourth-order dispersion. We present a solution which is accurate to the first order for such a perturbation. Our numerical simulations confirm the rogue wave existence when the parameter of perturbation |ν| < 0.05.

Wave Trains Generation in a Delayed Nonlinear Response of Bose-Einstein Condensates with Three-Body Interactions

We investigate the modulational instability of matter-wave condensates in a modified Gross-Pitaevskii equation which takes into account effects of the three-body interaction. This three-body interaction consists of a quintic term and an additional one representing the delayed nonlinear response of condensates which are trapped both in an attractive and a repulsive harmonic potentials. Our theoretical study uses a modified lens-type transformation and we obtain a modulational instability criterion, and an explicit growth rate. We show that the presence of the three-body interaction destabilizes the condensate, and enhances the appearance of instability in the condensate. Numerical experiments agree well with analytical predictions. Furthermore, our numerical simulations show that the three-body interaction modifies the symmetry of the trail of soliton chains created. The expulsive potential enhances the instability, while the attractive potential appears to soften the instability.

THREE-BODY INTERACTIONS BEYOND THE GROSS–PITAEVSKII EQUATION AND MODULATIONAL INSTABILITY OF BOSE–EINSTEIN CONDENSATES

Beyond the mean-field theory, a new model of the Gross–Pitaevskii equation (GPE) that describes the dynamics of Bose–Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the three-body interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.

Modulational instability of copropagating light beams induced by cubic–quintic nonlinearity in nonlinear negative-index material

We study the modulational instability (MI) of the continuous wave propagating through negative-index materials, based on the coupled cubic&#x2013;quintic nonlinear Schr&#xF6;dinger equations. A comparison is made with the already available results on MI without including the quintic term. We find that this term helps in achieving MI, in otherwise impossible combinations of dispersion and nonlinearity. For example, spatial MI can occur in the focusing regime, and temporal MI becomes possible in the anomalous dispersion for defocusing nonlinearity, while in the case of the normal dispersion regime, it occurs for focusing nonlinearity, and spatiotemporal MI can occur in the focusing nonlinearity and normal dispersion. Moreover, this additional term provides extra freedom to control the amplitude of the MI gain profile.

Suppressing the critical collapse of solitons by one-dimensional quintic nonlinear lattices

The stabilization of two-dimensional (2D) solitons in self-focusing Kerr media against the critical collapse with the help of nonlinear lattices (NLs), i.e., spatially periodic modulations of the local strength of the cubic nonlinearity, was recently investigated in detail. In one dimension (1D), the critical collapse is induced by the quintic self-focusing. Degenerate families of unstable localized modes which exist in these situations are usually called Townes solitons. We aim to explore a possibility of the stabilization of the 1D Townes solitons by means of NLs acting on the quintic or cubic–quintic (CQ) terms in the corresponding nonlinear-Schrödinger/Gross–Pitaevskii equation (NLSE/GPE). These settings may be realized in nonlinear optics and Bose–Einstein condensates (BECs). We develop the variational approximation (VA) for the CQ model, and use numerical methods to study the stability and mobility of solitons in both models, quintic and CQ. “Two-tier” and higher-order solitons, including dipole modes, are also found in the CQ medium (chiefly, in the case when the quintic nonlinearity tends to be self-defocusing). The stability region for fundamental solitons amounts to a narrow stripe in a parameter plane of the model with the quintic-only nonlinearity, being much broader in the case of the CQ nonlinearity.

Asymptotic stability of small gap solitons in nonlinear Dirac equations

We prove dispersive decay estimates for the one-dimensional Dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear Dirac equations with quintic and higher-order nonlinear terms.

Stabilization of one-dimensional solitons against the critical collapse by quintic nonlinear lattices

It has been recently discovered that stabilization of two-dimensional (2D) solitons against the critical collapse in media with the cubic nonlinearity by means of nonlinear lattices (NLs) is a challenging problem. We address the 1D version of the problem, i.e., the nonlinear-Schr\"odinger equation (NLSE) with the quintic or cubic-quintic (CQ) terms, the coefficient in front of which is periodically modulated in space. The models may be realized in optics and Bose-Einstein condensates (BECs). Stability diagrams for the solitons are produced by means of numerical methods and analytical approximations. It is found that the sinusoidal NL stabilzes solitons supported by the quintic-only nonlinearity in a narrow stripe in the respective parameter plane, on the contrary to the case of the cubic nonlinearity in 2D, where the stabilization of solitons by smooth spatial modulations is not possible at all. The stability region is much broader in the 1D CQ model, where higher-order solitons may be stable too.

Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation

We characterize the full family of soliton solutions sitting over a background plane wave and ruled by the cubic-quintic nonlinear Schroedinger equation in the regime where a quintic focusing term represents a saturation of the cubic defocusing nonlinearity. We discuss existence and properties of solitons in terms of catastrophe theory and fully characterize bistability and instabilities of the dark-antidark pairs, revealing new mechanisms of decay of antidark solitons.

Moving nonradiating kinks in nonlocalφ4andφ4-φ6models

We explore the existence of moving nonradiating kinks in nonlocal generalizations of φ(4) and φ(4)-φ(6) models. These models are described by nonlocal nonlinear Klein-Gordon equation, u(tt)-Lu+F(u)=0, where L is a Fourier multiplier operator of a specific form and F(u) includes either just a cubic term (φ(4) case) or cubic and quintic (φ(4)-φ(6) case) terms. The general mechanism responsible for the discretization of kink velocities in the nonlocal model is discussed. We report numerical results obtained for these models. It is shown that, contrary to the traditional φ(4) model, the nonlocal φ(4) model does not admit moving nonradiating kinks but admits solitary waves that do not exist in the local model. At the same time the nonlocal φ(4)-φ(6) model describes moving nonradiating kinks. The set of velocities allowed for these kinks is discrete with the highest possible velocity c(1). This set of velocities is unambiguously determined by the parameters of the model. Numerical simulations show that a kink launched at the velocity c higher than c(1) starts to decelerate, and its velocity settles down to the highest value of the discrete spectrum c(1).

Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas

The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential -1/r^2), or "quantum anomaly", is a well-known issue in the quantum theory. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem different from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge, and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schroedinger equation (SE). The addition of the harmonic trap gives rise to a tristability, in the case when the SE still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counter-intuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential 1/r^2. In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential 1/r^2 gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form, and also in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.

Bifurcation of gap solitons in periodic potentials with a periodic sign-varying nonlinearity coefficient

We address the Gross–Pitaevskii equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works, we show that the intersite cubic nonlinear terms in the discrete nonlinear Schrödinger (DNLS) equation appear beyond the applicability of assumptions of the tight-binding approximation. Instead of these terms, for an even linear potential and an odd nonlinearity coefficient, the DNLS equation and other reduced equations have the quintic nonlinear term, which correctly describes bifurcation of gap solitons in the semi-infinite gap.

Modulational instability and exact solutions of the discrete cubic–quintic Ginzburg–Landau equation

In this paper, we investigate analytically and numerically the modulational instability in a model of nonlinear physical systems such as Bose–Einstein condensates in a deep optical lattice. This model is described by the discrete complex cubic–quintic Ginzburg–Landau equation with a non-local quintic term. We obtain characteristics of the modulational instability in the form of typical dependences of the instability growth rate (gain) on the perturbation wavenumber and the system's parameters. Excellent agreement has been obtained between the analytical and numerical study. Further, we derive the periodic function and new type of solitary wave solutions for the above system. By using the extended Jacobi elliptic function approach, we obtain the exact stationary solitons and periodic wave solutions of this equation. These solutions include the Jacobi periodic wave solution, alternating phase Jacobi periodic wave solution, kink and bubble soliton solutions, alternating phase kink soliton and alternating phase bubble soliton solutions.

An accurate and precise polynomial model of angular interrogation surface plasmon resonance data

We present a simple, statistically based method of fitting waveguide-coupled surface plasmon resonance (WCSPR) angular interrogation experiment data in the vicinity of the resonance angle using an appropriate polynomial model. This method allows one to determine the resonance angle to within precision of as little as 2% of the sampling step size, with mean results averaging about 8% of the step size, better than an order of magnitude improvement over no regression, achieved with little effort. In testing this method, we use theoretical and experimental WCSPR data. We have compared the statistical significance of using additional terms in a given polynomial representation. F-Ratio tests based on the “extra sum of squares” principle indicate that, in the vicinity of the resonance, approximately 20 millidegrees about the minimum, the addition of quintic or higher order terms to the quartic polynomial representation is not statistically significant. We have found that both cubic and quartic models produce estimates of the position of the minimum in which the confidence interval is both accurate and precise with an error of less than one tenth of a millidegree. In addition, a similar analysis of theoretical calculations suggests that this polynomial method, which is generally applicable to the determination of extrema in any spectrum, is capable of very high accuracy and precision.

Abundant exact solutions for the higher order non-linear Schrödinger equation with cubic–quintic non-Kerr terms

In this paper, the higher order NLS equation with cubic–quintic non-linear terms arising in non-Kerr media is studied, new abundant solitary solutions of this equation are obtained using generalized auxiliary equation method.

Stable vortex solitons in the Ginzburg-Landau model of a two-dimensional lasing medium with a transverse grating

We introduce a two-dimensional model of a laser cavity based on the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity and a lattice potential accounting for the transverse grating. A remarkable fact is that localized vortices, built as sets of four peaks pinned to the periodic potential, may be stable without the unphysical diffusion term, which was necessary for the stabilization in previously studied models. The vortices are chiefly considered in the onsite (rhombic) form, but the stabilization of offsite vortices (square-shaped ones) and quadrupoles is demonstrated too. Stability regions for the rhombic vortices and fundamental solitons are identified in the model's parameter space, and scenarios of the evolution of unstable vortices are described. An essential result is a minimum strength of the lattice potential which is necessary to stabilize the vortices. The stability border is also identified in the case of the self-focusing quintic term in the underlying model, which suggests a possibility of the supercritical collapse. Beyond this border, the stationary vortex turns into a vortical breather, which is subsequently replaced by a dipolar breather and eventually by a single-peak breather.

Stabilization and destabilization of second-order solitons against perturbations in the nonlinear Schrödinger equation

We consider splitting and stabilization of second-order solitons (2-soliton breathers) in a model based on the nonlinear Schrödinger equation, which includes a small quintic term, and weak resonant nonlinearity management (NLM), i.e., time-periodic modulation of the cubic coefficient, at the frequency close to that of shape oscillations of the 2-soliton. The model applies to the light propagation in media with cubic-quintic optical nonlinearities and periodic alternation of linear loss and gain and to Bose–Einstein condensates, with the self-focusing quintic term accounting for the weak deviation of the dynamics from one dimensionality, while the NLM can be induced by means of the Feshbach resonance. We propose an explanation to the effect of the resonant splitting of the 2-soliton under the action of the NLM. Then, using systematic simulations and an analytical approach, we conclude that the weak quintic nonlinearity with the self-focusing sign stabilizes the 2-soliton, while the self-defocusing quintic nonlinearity accelerates its splitting. It is also shown that the quintic term with the self-defocusing/focusing sign makes the resonant response of the 2-soliton to the NLM essentially broader in terms of the frequency.

Investigation on Dynamics of the Extended Duffing-Van der Pol System

Abstract The chaotic motion in periodic self-excited oscillators has been extensively investigated through experiments and computer simulations. However, with the advent of the study of chaotic motion by means of strange attractors, Poincar´e map, fractal dimension, it has become necessary to seek for a better understanding of nonlinear system with higher order nonlinear terms. In this paper we consider an extended Duffing-Van der Pol oscillator by introducing a nonlinear quintic term. The dynamical behaviour of the system is investigated by using Melnikov analysis and numerical simulation. The results can help one to understand the essence of given nonlinear system.

Symmetric and asymmetric bound states for the nonlinear Schrödinger equation with inhomogeneous nonlinearity

We introduce a model of a Bose–Einstein condensate based on the one-dimensional nonlinear Schrödinger equation, in which the nonlinear term depends on the domain. The nonlinear term changes a cubic term into a quintic term, according to the domain considered. We study the existence, stability and bifurcation of solutions, and use the qualitative theory of dynamical systems to study certain properties of such solutions.

Collective Excitations and Nonlinear Dynamics of 1D BEC with Two- and Three-body Interactions in Anharmonic Traps

The collective excitations of low-dimensional Bose–Einstein condensates with two- and three-body interactions in anharmonic potentials are investigated. Using the standard variational approach, the governing equations of motions for the low-energy excitations are obtained by solving time-dependent Gross–Pitaevskii–Ginzburg equation, and the excitation spectrums are calculated in small amplitude limit. The frequency shift and nonlinear mode coupling induced by the anharmonic distortion (adding cubic, quartic, or quintic term to a harmonic trap) are studied.