Quintic Terms Research Articles

Solitons, breathers and rogue waves for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in an erbium-doped fiber system

In this paper, a higher-order nonlinear Schrödinger–Maxwell–Bloch system with quintic terms is investigated, which describes the propagation of ultrashort optical pulses, up to the attosecond duration, in an erbium-doped fiber. Multi-soliton, breather and rogue-wave solutions are derived by virtue of the Darboux transformation and the limiting procedure. Features and interaction patterns of the solitons, breathers and rogue waves are discussed. (i) The solitonic amplitudes, widths and velocities are exhibited, and solitonic amplitudes and widths are proved to have nothing to do with the higher-order terms. (ii) The higher-order terms and frequency detuning affect the growth rate of periodic modulation and skewing angle for the breathers, except for the range of the frequency of modulation. (iii) The quintic terms and frequency detuning have the effects on the temporal duration for the rogue waves. (iv) Breathers are classified into two types, according to the range of the modulation instability. (v) Interaction between the two solitons is elastic. When the two solitons interact with each other, the periodic structure occurs, which is affected by the higher-order terms and frequency detuning. (vi) Interaction between the two Akhmediev-like breathers or two Kuznetsov–Ma-like solitons shows the different patterns with different ratios of the relative modulation frequencies, while the interaction area induced by the two breathers looks like a higher-order rogue wave.

Modulational instability in nonlocal media with competing non-Kerr nonlinearities

We investigate analytically and numerically the modulational instability (MI) and propagation properties of light in nonlocal media with competing cubic–quintic nonlinearities where the response functions are assumed to be equal. By using the linear stability analysis, the generic properties of the MI gain spectra are demonstrated for the exponential and rectangular response functions. Special attention is paid to investigate the competition between the spatial scale of the cubic and quintic nonlinearities. For media with exponential response function, we have obtained the range of the wave numbers where instability occurs. It is found that the increase in the absolute value of the quintic nonlinearity suppresses the instability in the regime where the cubic nonlinearity prevails over the quintic one and promotes its development in the opposite case. For media with negative response function, additional MI bands are excited at higher wave numbers when the width of the nonlocal response function exceeds a certain threshold. In the regime where the quintic nonlinearity is dominant, the increase in the absolute value of the quintic coefficient leads to the enhancement of the gain value and the movement of the maximum gain to higher wave numbers. On the other hand, in the case of the predominance of the cubic nonlinearity, the position of the maximum gain bands move to lower wave numbers and MI domain becomes increasingly narrows when the quintic term increases. The numerical simulations fully confirm our analytical results.

Transverse instability of solitons in nonlinear systems

We study the transverse instability associated with higher-dimensional nonlinear systems. For this, a cubic–quintic-nonlinear system is considered wherein the coefficients of cubic and quintic terms are allowed to be of either focusing or defocusing nature. It is found that focusing-cubic and defocusing-quintic nonlinearity can suppress the transverse instability growth rate. The addition of guiding potentials can also suppress the instability growth rate for all kinds of solitons with the same propagation constant.

Higher-Order Rogue Waves for a Fifth-Order Dispersive Nonlinear Schrödinger Equation in an Optical Fibre

Abstract In this article, a fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort optical pulses, up to the attosecond duration, in an optical fibre. Rogue wave solutions are derived by virtue of the generalised Darboux transformation. Rogue wave structures and interaction are discussed through (i) the analyses on the higher-order rogue waves, the cubic, quartic, quintic, group-velocity, and phase-parameter effects; (ii) a higher-order rogue wave consisting of the first-order rogue waves via the interaction; (iii) characteristics of the rogue waves which are summarised, including the maximum/minimum values of the rogue waves and the number of the first-order rogue waves for composing the higher-order rogue wave; and (iv) spatial-temporal patterns which are illustrated and compared with those of the ‘self-focusing’ nonlinear Schrödinger equation. We find that the quintic terms increase the time of appearance for the first-order rogue waves which form the higher-order rogue wave, and that the quintic terms affect the interaction among the first-order rogue waves, which elongates the distance of appearance for the higher-order rogue wave.

Rogue wave solutions of a higher-order Chen–Lee–Liu equation

We consider a next-higher-order extension of the Chen–Lee–Liu equation, i.e., a higher-order Chen–Lee–Liu (HOCLL) equation with third-order dispersion and quintic nonlinearity terms. We construct the n-fold Darboux transformation (DT) of the HOCLL equation in terms of the n × n determinants. Comparing this with the nonlinear Schrödinger equation, the determinant representation Tn of this equation is involved with the complicated integrals, although we eliminate these integrals in the final form of the DT, so that the DT of the HOCLL equation is unusual. We provide explicit expressions of multi-rogue wave (RW) solutions for the HOCLL equation. It is concluded that the rogue wave solutions are likely to be crucial when considering higher-order nonlinear effects.

Breaking of Vainshtein screening in scalar-tensor theories beyond Horndeski

The Horndeski theory of gravity is known as the most general scalar-tensor theory with second-order field equations. Recently, it was demonstrated by Gleyzes et al. that the Horndeski theory can further be generalized in such a way that although field equations are of third order, the number of propagating degrees of freedom remains the same. We study small-scale gravity in the generalized Horndeski theory, focusing in particular on an impact of the new derivative interaction beyond Horndeski on the Vainshtein screening mechanism. In the absence of the quintic Galileon term and its generalization, we show that the new interaction does not change the qualitative behavior of gravity outside and near the source: the two metric potentials coincide, $\mathrm{\ensuremath{\Phi}}=\mathrm{\ensuremath{\Psi}}(\ensuremath{\sim}{r}^{\ensuremath{-}1})$, while the gravitational coupling is given by the cosmological one and hence is time dependent in general. We find, however, that the gravitational field inside the source shows a novel behavior due to the interaction beyond Horndeski: the gravitational attraction is not determined solely from the enclosed mass and two potentials do not coincide, indicating breaking of the screening mechanism.

Effect of the amplitude of vibrations on the pull-in instability of double-sided actuated microswitch resonators

This paper exhibits the effect of the amplitude of vibrations on the pull-in instability and nonlinear natural frequency of a double-sided actuated microswitch by using a nonlinear frequency-amplitude relationship. The nonlinear governing equation of the microswitch pre-deformed by an electric field includes even and odd nonlinearities with a quintic nonlinear term. The study is performed by a new analytical method called the Hamiltonian approach (HA). It is demonstrated that the first term in series expansions is sufficient to produce an acceptable solution. Results obtained by numerical methods validate the soundness of the asymptotic procedure.

Dynamic pull-in instability of double-sided actuated nano-torsional switches

A nonlinear frequency-amplitude relation is developed to investigate the vibrational amplitude effect on the dynamic pull-in instability of double-sided-actuated nano-torsional switches. The governing equation of a nano-electro-mechanical system pre-deformed by an electric field contains the quintic nonlinear term. The influences of basic parameters on the pull-in instability and natural frequency are investigated using a powerful analytical approach called the homotopy perturbation method. It is demonstrated that two terms in series expansion are sufficient to produce an acceptable solution. The numerical results obtained have verified the soundness of the asymptotic procedure. The phase portraits of the double-sided nano-torsional-actuator exhibit periodic, homoclinic and heteroclinic orbits.

Properties of Optical Soliton in a Three Level Medium with Quintic Nonlinearity

Propagation characteristics of optical soliton in a three level atomic medium are analyzed by treating the material medium quantum mechanically, but the electromagnetic wave classically. Both the cubic and quintic components of the nonlinear polarization of the electromagnetic field are considered along with those generated due to the dipole formation of the material. A numerical simulation is carried out with the help of split-step technique. It is observed that the power of the pulse, distance of propagation and degree of dispersion are intimately related. The role of polarization due to the material is duely compensated by keeping higher order dispersive terms. In this connection we have seen that keeping the higher order dispersive terms, up to the eighth order, which is actually the phenomenon of continuum generation, results in a better form of the pulse. In our paper, we have analyzed the effects in both the cases, that is, including and excluding the quintic terms and in each case we have considered the effects of second-order dispersion (β2) as well as the higher order dispersion terms ( 8 2 j j β =

Stabilization of solitons under competing nonlinearities by external potentials.

We report results of the analysis for families of one-dimensional (1D) trapped solitons, created by competing self-focusing (SF) quintic and self-defocusing (SDF) cubic nonlinear terms. Two trapping potentials are considered, the harmonic-oscillator (HO) and delta-functional ones. The models apply to optical solitons in colloidal waveguides and other photonic media, and to matter-wave solitons in Bose-Einstein condensates loaded into a quasi-1D trap. For the HO potential, the results are obtained in an approximate form, using the variational and Thomas-Fermi approximations, and in a full numerical form, including the ground state and the first antisymmetric excited one. For the delta-functional attractive potential, the results are produced in a fully analytical form, and verified by means of numerical methods. Both exponentially localized solitons and weakly localized trapped modes are found for the delta-functional potential. The most essential conclusions concern the applicability of competing Vakhitov-Kolokolov (VK) and anti-VK criteria to the identification of the stability of solitons created under the action of the competing SF and SDF terms.

Domain walls and bubble droplets in immiscible binary Bose gases

The existence and stability of domain walls (DWs) and bubble-droplet (BD) states in binary mixtures of quasi-one-dimensional ultracold Bose gases with inter- and intra-species repulsive interactions is considered. Previously, DWs were studied by means of coupled systems of Gross-Pitaevskii equations (GPEs) with cubic terms, which model immiscible binary Bose-Einstein condensates (BECs). We address immiscible BECs with two- and three-body repulsive interactions, as well as binary Tonks--Girardeau (TG) gases, using systems of GPEs with cubic and quintic nonlinearities for the binary BEC, and coupled nonlinear Schr\"{o}dinger equations with quintic terms for the TG gases. Exact DW\ solutions are found for the symmetric BEC mixture, with equal intra-species scattering lengths. Stable asymmetric DWs in the BEC mixtures with dissimilar interactions in the two components, as well as of symmetric and asymmetric DWs in the binary TG gas, are found by means of numerical and approximate analytical methods. In the BEC system, DWs can be easily put in motion by phase imprinting. Combining a DW and anti-DW on a ring, we construct BD states for both the BEC and TG models. These consist of a dark soliton in one component (the "bubble"), and a bright soliton (the "droplet") in the other. In the BEC system, these composite states are mobile too.

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

Solitary wave solutions and modulational instability analysis of the nonlinear Schrödinger equation with higher-order nonlinear terms in the left-handed nonlinear transmission lines

We report the modulational instability (MI) analysis for the modulation equations governing the propagation of modulated waves in a practical left-handed nonlinear transmission lines with series of nonlinear capacitance. Considering the voltage in the spectral domain and the Taylor series around a certain modulation frequency, we show in the continuum limit, that the dynamics of localized signals is described by a nonlinear Schrödinger equation with a cubic–quintic nonlinear terms. The MI process is then examined and we derive the gain spectra of MI for the generation of solitonlike-object in the transmission line metamaterials. We emphasize on the effect of losses on the MI gain spectra. An exact kink-darklike solutions is derived through the auxiliary equation method. It comes out that the width of the darklike solution decreases as the attenuation constant increases. Our theoretical solution is in good agreement with our numerical observation.

Spatial control of the competition between self-focusing and self-defocusing nonlinearities in one- and two-dimensional systems

We introduce a system with competing self-focusing (SF) and self-defocusing (SDF) terms, which have the same scaling dimension. In the one-dimensional (1D) system, this setting is provided by a combination of the SF cubic term multiplied by the delta-function, $\delta (x)$, and a spatially uniform SDF quintic term. This system gives rise to the most general family of 1D-Townes solitons, the entire family being unstable. However, it is completely stabilized by a finite-width regularization of the $\delta $-function. The results are produced by means of numerical and analytical methods. We also consider the system with a symmetric pair of regularized $\delta $-functions, which gives rise to a wealth of symmetric, antisymmetric, and asymmetric solitons, linked by a bifurcation loop, that accounts for the breaking and restoration of the symmetry. Soliton families in 2D versions of both the single- and double-delta-functional systems are also studied. The 1D and 2D settings may be realized for spatial solitons in optics, and in Bose-Einstein condensates.

Higher-order rogue wave solutions of the Kundu–Eckhaus equation

In this paper, we investigate higher-order rogue wave solutions of the Kundu–Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu–Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu–Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu–Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged.

The rogue waves with quintic nonlinearity and nonlinear dispersion effects in nonlinear optical fibers

We study on rogue waves in a nonlinear fiber with cubic and quintic nonlinearities, linear and nonlinear dispersion effects. We find the rogue wave with these higher order effects has identical shape with the well-known one in nonlinear Schro¨dinger equation. However, the quintic nonlinear term and nonlinear dispersion effect affect the velocity of rogue wave, and the evolution of its phase.

Cubic–quintic long-range interactions with double well potentials

In the present work, we examine the combined effects of cubic and quintic terms of the long-range type in the dynamics of a double well potential. Employing a two-mode approximation, we systematically develop two cubic–quintic ordinary differential equations and assess the contributions of the long-range interactions in each of the relevant prefactors, gauging how to simplify the ensuing dynamical system. Finally, we obtain a reduced canonical description for the conjugate variables of relative population imbalance and relative phase between the two wells and proceed to a dynamical systems analysis of the resulting pair of ordinary differential equations. While in the case of cubic and quintic interactions of the same kind (e.g. both attractive or both repulsive), only a symmetry-breaking bifurcation can be identified, a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions is a ‘symmetry-restoring’ bifurcation. Namely, in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the antisymmetric state, there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the antisymmetric parent one. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. The model is argued to be of physical relevance, especially so in the context of optical thermal media.

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity

We investigate mobility regimes for localized modes in the discrete nonlinear Schrödinger (DNLS) equation with the cubic-quintic on-site terms. Using the variational approximation, the largest soliton's total power admitting progressive motion of kicked discrete solitons is predicted by comparing the effective kinetic energy with the respective Peierls-Nabarro (PN) potential barrier. The prediction, for the DNLS model with the cubic-only nonlinearity too, demonstrates a reasonable agreement with numerical findings. A small self-focusing quintic term quickly suppresses the mobility. In the case of the competition between the cubic self-focusing and quintic self-defocusing terms, we identify parameter regions where odd and even fundamental modes exchange their stability, involving intermediate asymmetric modes. In this case, stable solitons can be set in motion by kicking, so as to let them pass the PN barrier. Unstable solitons spontaneously start oscillatory or progressive motion, if they are located, respectively, below or above a mobility threshold. Collisions between moving discrete solitons, at the competing nonlinearities frame, are studied too.

Multipole solitary wave solutions of the higher-order nonlinear Schrödinger equation with quintic non-Kerr terms

We consider a high-order nonlinear Schrödinger (HNLS) equation with third- and fourth-order dispersions, quintic non-Kerr terms, self steepening, and self-frequency-shift effects. The model applies to the description of ultrashort optical pulse propagation in highly nonlinear media. We propose a complex envelope function ansatz composed of single bright, single dark and the product of bright and dark solitary waves that allows us to obtain analytically different shapes of solitary wave solutions. Parametric conditions for the existence and uniqueness of such solitary waves are presented. The solutions comprise fundamental solitons, kink and anti-kink solitons, W-shaped, dipole, tripole, and fifth-order solitons. In addition, we found a new type of solitary wave solution that takes the shape of N, illustrating the potentially rich set of solitary wave solutions of the HNLS equation. Finally, the stability of the solutions is checked by direct numerical simulation.