Quintic Nonlinearity Research Articles

Soliton shedding from Airy pulses in a highly dispersive and nonlinear medium

We present a numerical investigation of the propagation dynamics of a truncated Airy pulse in a highly dispersive and nonlinear medium by employing the split-step Fourier transform method and look, in particular, into the effects of fourth-order dispersion (FOD) and cubic-quintic-septic nonlinearity on pulse evolution. Presence of FOD cancels the Airy pulse’s self-acceleration along with eclipsing the oscillatory tail during propagation in the linear regime. Further, we observe soliton shedding at low input pulse power in the presence of cubic and quintic nonlinearity and negative FOD. The emergent soliton exhibits temporal shift, and the direction and the extent of the shift depend upon the strengths of cubic and quintic nonlinearities. In the presence of anomalous group-velocity dispersion (GVD) with negative FOD, soliton shedding is observed at relatively high input pulse power. The strengths of GVD and nonlinearity play a vital role in the temporal shifting of the emergent soliton. Furthermore, we have explored the effects of septic nonlinearity on soliton shedding in different scenarios of nonlinearity and dispersion.

Impact of nonlocality and quintic local nonlinearity on the pulses interaction and the dynamical characteristics parameters of soliton in weakly nonlinear nonlocal media

We study the impact of nonlocality and local quintic nonlinearity on the dynamic characteristic parameters of spatial optical bright soliton and on their interaction and collision in weakly nonlocal nonlinear media. For this purpose, we establish the propagation equation governing this dynamic which is a nonlinear cubic-quintic Schrödinger equation. We approach the solution of this equation by a secant hyperbolic ansatz. By the means of the variational method, we bring out the action mode of the nonlocality and quintic nonlinearity on the pulse parameters. Subsequently, a numerical analysis is made and allows us to conclude on their role on the dynamic of pulse in these media. It follows that quintic nonlinearity contributes to the pulse compression and the increase of phase-front curvature, while nonlocality participates in enlargement and the decrease of the phase-front curvature. Taking into account the two effects helps stabilize the soliton during its propagation by generating a new dynamic state depending on whether the quintic nonlinearity and the nonlocality add up, compensate each other, or that one of the two effects is more important than the other. Concerning their interaction, we showed that quintic nonlinearity provides a repulsive force when nonlocality provides an attractive force. In the case of collision, quintic nonlinearity can induce elastic or inelastic collision.

Higher-order dispersion and nonlinear effects of optical fibers under septic self-steepening and self-frequency shift.

We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schrödinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

Darboux transformation, generalized Darboux transformation and vector breather solutions for a coupled variable-coefficient cubic-quintic nonlinear Schrödinger system in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide

Non-Kerr media, twin-core nonlinear optical fibers and waveguides are widely applied in optical fiber communications. In order to model the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide, a coupled variable-coefficient cubic-quintic nonlinear Schrödinger system is investigated in this paper. For the two components of the electromagnetic fields, on the basis of the existing Lax pair, we take a gauge transformation to convert a Lax pair into an Ablowitz-Kaup-Newell-Segur system. Then we acquire a system, which is equal to the original system. The N-fold Darboux transformation is derived based on our Lax pair, where N is a positive integer. Applying Taylor series, we derive the N-fold generalized Darboux transformation. The second- and third-order vector breather solutions are obtained via the generalized Darboux transformation method. All our results depend on the variable coefficients of the original system. Our results might be of some use in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide system.

Local well-posedness of a generalized nonlinear Schrödinger equation with cubic–quintic nonlinearity

In this article, we study the Cauchy Problem associated to the generalized cubic–quintic Schrödinger equation which has been used for the description of propagation pulses in optical fibers. Using gauge transforms and a derivation process, we prove under some regularity assumptions, the local well-posedness. Additionally, a global H1 bound is established for the solution.

Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear dispersive equations via deep learning

In the field of mathematical physics, there exist many physically interesting nonlinear dispersive equations with peakon solutions, which are solitary waves with discontinuous first-order derivative at the wave peak. In this paper, we apply the multi-layer physics-informed neural networks (PINNs) deep learning to successfully study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations with initial–boundary value conditions such as the Camassa–Holm (CH) equation, Degasperis–Procesi equation, modified CH equation with cubic nonlinearity, Novikov equation with cubic nonlinearity, mCH-Novikov equation, b-family equation with quartic nonlinearity, generalized modified CH equation with quintic nonlinearity, and etc. Moreover, we also study the data-driven parameter discovery of the CH equation with the aid of the single peakon These results will be useful to further study the peakon solutions and corresponding experimental design of nonlinear dispersive equations.

Effects of giant Kerr and quintic nonlinearities on electromagnetically induced grating in multiple quantum wells

Effects of giant Kerr and quintic nonlinearities on electromagnetically induced grating in multiple quantum wells

Supercontinuum generation of truncated Airy pulses in a cubic-quintic AsSe2/As2S5 optical waveguide with rib-like structure

In this paper, we report a numerical study of supercontinuum generation (SCG) using specially truncated Airy pulses in the mid-infrared (MIR) region of wavelengths through an AsSe2/As2S5 optical waveguide with rib-like structures modeled by the cubic-quintic nonlinear Schrödinger equation (CQNLSE). We use the waveguide studied by Diouf et al (2019 J. Lightwave Technol. 37 5692), which has been found interesting, as it can generate explosive spectra in the MIR domain. Through the anomalous dispersion regime, the spectral 20 dB bandwidths obtained have been found to exceed approximately 97250 nm. We also consider the effects on SCG spectra of quintic Kerr nonlinearity (QKN) and nonlinear photon absorptions (NPAs) such as two-photon absorption (TPA) and three-photon absorption (3PA). Without the NPAs, only the cooperative QKN is beneficial for the MIR-SCG. In the temporal domain, the emission of dispersive waves (DWs) is avoided by the presence of NPAs, while the cooperative nonlinearities perform this emission. Furthermore, if, in the cooperative case, we always neglect the 3PA and consider the TPA, we find an interesting spectral intensity (SI) of the SCG, compared with the case of competition between nonlinearities. Surprisingly, the 3PA appears to be a good tool for controlling the TPA because the SI is less reduced when the QKN cooperates with the cubic Kerr nonlinearity (CKN). In particular, if we consider the case of competition, the 3PA enhances the SI compared to the single TPA case (in which the 3PA is neglected) for larger wavelengths.

Phase engineering of chirped rogue waves in Bose–Einstein condensates with a variable scattering length in an expulsive potential

We consider a cubic Gross-Pitaevskii (GP) equation governing the dynamics of Bose-Einstein condensates (BECs) with time-dependent coefficients in front of the cubic term and inverted parabolic potential. Under a special condition imposed on the coefficients, a combination of phase-imprint and modified lens-type transformations converts the GP equation into the integrable Kundu-Eckhaus (KE) one with constant coefficients, which contains the quintic nonlinearity and the Raman-like term producing the self-frequency shift. The condition for the baseband modulational instability of CW states is derived, providing the possibility of generation of chirped rogue waves (RWs) in the underlying matter-wave (BEC) model. Using known RW solutions of the KE equation, we present explicit first- and second-order chirped RW states. The chirp of the first- and second-order RWs is independent of the phase imprint. Detailed solutions are presented for the following configurations: (i) the nonlinearity exponentially varying in time; (ii) time-periodic modulation of the nonlinearity; (iii) a stepwise time modulation of the strength of the expulsive potential. Singularities of the local chirp coincide with valleys of the corresponding RWs. The results demonstrate that the temporal modulation of the s-wave scattering length and strength of the inverted parabolic potential can be used to manipulate the evolution of rogue matter waves in BEC.

The [formula omitted]-fold degenerate Darboux transformation of the three component Kundu–Eckhaus equations: Novel interactions between soliton and positon solution

In this paper, an explicit representation of the N-fold degenerate Darboux transformation (DT) for the three component Kundu–Eckhaus equations, which can be viewed as a model to describe the effect of quintic nonlinearity on the ultra-short optical pulse propagation in non-Kerr media, has been derived. The DT is expressed by the initial eigenfunctions, spectral parameters and ‘seed’ solution. As applications of DT, the abundant and novel interactions between soliton and positon are discussed in detail, such as elastic interactions, inelastic interactions and bound states.

Dissipative structures in the resonant interaction of laser radiation with nonlinear dispersive medium

The article presents the results of studies on the stability of dissipative structures (DS) arising in the resonant interaction of laser radiation with a nonlinear medium. Resonant interaction is modeled by the one dimensional complex Ginzburg-Landau equation with a nonconservative cubic–quintic nonlinearity. The areas of existence of stable DS solutions have been determined analytically using a variational approach and confirmed numerically by extensive numerical simulations.

Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential

We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schrödinger equation with cubic or cubic-quintic (CQ) nonlinearity and a parity-time-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of fractional diffraction. While GSs are not true solutions, direct simulations confirm that their shapes and results of their stability analysis provide a "blueprint" for the evolution of genuine localized modes in the system.

Akbari–Ganji Method for Solving Equations of Euler–Bernoulli Beam with Quintic Nonlinearity

In many real word applications, beam has nonlinear transversely vibrations. Solving nonlinear beam systems is complicated because of the high dependency of the system variables and boundary conditions. It is important to have an accurate parametric analysis for understanding the nonlinear vibration characteristics. This paper presents an approximate solution of a nonlinear transversely vibrating beam with odd and even nonlinear terms using the Akbari–Ganji Method (AGM). This method is an effective approach to solve nonlinear differential equations. AGM is already used in the heat transfer science for solving differential equations, and in this research for the first time, it is applied to find the approximate solution of a nonlinear transversely vibrating beam. The advantage of creating new boundary conditions in this method in additional to predefined boundary conditions is checked for the proposed nonlinear case. To illustrate the applicability and accuracy of the AGM, the governing equation of transversely vibrating nonlinear beams is treated with different initial conditions. Since simply supported and clamped-clamped structures can be encountered in many engineering applications, these two boundary conditions are considered. The periodic response curves and the natural frequency are obtained by AGM and contrasted with the energy balance method (EBM) and the numerical solution. The results show that the present method has excellent agreements in contrast with numerical and EBM calculations. In most cases, AGM is applied straightforwardly to obtain the nonlinear frequency– amplitude relationship for dynamic behaviour of vibrating beams. The natural frequencies tested for various values of amplitude are clearly stated the AGM is an applicable method for the proposed nonlinear system. It is demonstrated that this technique saves computational time without compromising the accuracy of the solution. This approach can be easily extended to other nonlinear systems and is therefore widely applicable in engineering and other sciences.

Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction

We study low regularity behavior of the nonlinear wave equation in $\mathbb {R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb {R}^2)\times H^{s-1}(\mathbb {R}^2)$ is ill-posed whenever $0 < s < s_{cr}$, where the critical exponent $s_{cr}$ depends on the degree of nonlinearity. In particular, for the quintic nonlinearity $u^5$, the critical exponent in $\mathbb {R}^2$ is $s_{cr} = 1/2$, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents $s$ such that $-1/6 < s \le s_{cr} = 1/2$. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.

Propagation of chirped periodic and localized waves with higher-order effects through optical fibers

We study the propagation properties of nonlinear periodic waves in an optical fiber medium exhibiting Kerr dispersion and quintic nonlinearity. The quintic derivative nonlinear Schrödinger equation is applied to model the evolution of femtosecond light waves in such system. Unlike periodic structures in Kerr media, the novel waves possess a nonlinear chirp which varies with their intensity. In addition to the linear part, the pulse chirp also includes two intensity dependent chirping terms. The conditions on fiber parameters for the existence of the newly found periodic waves are presented. A wide variety of solitary pulse shapes including the bright, dark, kink, anti-kink, and gray solitary waves are obtained in the long-wave limit. Our results show that Kerr dispersion plays a crucial role in introducing a nonlinear chirp for the periodic and solitary waves. Finally, the stability of these nonlinearly chirped solutions is numerically studied under the finite perturbations.

Phase-sensitive modulation instability in asymmetric coupled quantum wells

This paper presents a theoretical investigation of the phase-sensitive modulation instability of a continuous or quasicontinuous probe beam in an asymmetric coupled quantum well (ACQW) system facilitated by electromagnetically induced transparency. The dispersive properties of the probe beam can be controlled not only by the Rabi frequency of the control field but by the relative phase between them as well. The ACQW system exhibits large Kerr and quintic nonlinearities which could be controlled by controlling the relative phase of the probe beam. The probe beam is modulation unstable, and the gain and the bandwidth of unstable frequencies could be controlled by the phase of the probe beam. The gain of the instability disappears at certain values of phase and at certain other values of the phase the gain of the instability, and the bandwidth of unstable frequencies could be made large. Similarly, at certain values of the probe phase, the interplay of fourth-order dispersion and nonlinearity leads to the creation of discrete sidebands of the modulation instability. Both the fourth-order dispersion and quintic nonlinearity considerably reduces the growth and bandwidth of unstable frequencies.

Modulation instability induced supercontinuum generation in liquid core suspended photonic crystal fiber with cubic-quintic nonlinearities

In this paper, we propose a liquid core suspended photonic crystal fiber (LCSPCF) as a potential waveguide structure for nonlinear applications. We emphasize the dramatic improvement of the nonlinear properties of the PCF through the infiltration of liquid and highlight the effect of suspension in the further enhancement of nonlinearity. To demonstrate the capability of LCSPCF, we perform a modulation instability (MI) study and illustrate the MI enhancement in the LCSPCF against conventional PCF. We observe that the quintic nonlinearity (QN) increases the MI for cooperating nonlinearities while it suppresses the MI for competing nonlinearities. We have also importantly studied the effect of QN on the MI induced supercontinuum generation (SCG) for silica and CS2 liquid core PCF. Thus, we propose the LCSPCF as a platform for nonlinear applications such as SCG, pulse compression, etc., owing to the presence of high and controllable nonlinear coefficients.

Stability of moving Bragg solitons in a semilinear coupled system with cubic–quintic nonlinearity

We investigate the photonic bandgap structure, soliton properties and their stability in a semilinear coupled system where one core has cubic–quintic nonlinearity and is equipped with a Bragg grating, while the other one is uniform and linear. The investigation reveals that three distinct bandgaps, namely the upper, lower and central gaps, exist in the model. It is shown that the model supports two disjoint soliton families which are designated as Type 1 and Type 2 solitons. The soliton families exist only in the upper and lower bandgaps and are separated by a boundary at which no soliton solutions exist. A systematic numerical stability analysis is performed. It is found that vast regions in both the upper and lower bandgaps exist where Type 1 solitons are stable. However, Type 2 solitons are found to be always unstable. The effects of system parameters such as group velocity mismatch between the cores, velocity of solitons, and the coupling coefficient on the stability of solitons are analysed.

Infinite energy solutions for weakly damped quintic wave equations in ℝ³

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.

Image encryption using a novel quintic jerk circuit with adjustable symmetry

SummaryAn electronic implementation of a novel chaotic oscillator with quintic nonlinearity is proposed herein. Dynamical behaviors of the system are investigated using well‐known numerical simulations and analyses such as phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The chaotic circuit presents an inversion‐symmetry, and we show that it can exhibit some nonlinear phenomena specific to symmetric systems, such as symmetric bifurcations, symmetric attractors, coexisting symmetric bubbles, and coexisting symmetric attractors. Since symmetry is never perfect, some symmetry imperfections must be always assumed to be present. Thus, an external Direct Current (DC) voltage is introduced in order to highlight the influence of asymmetry on the dynamics of the chaotic oscillator. It is found that more complex nonlinear behaviors occur in the presence of symmetry breaking like asymmetric coexisting bifurcations, asymmetric attractors, coexisting asymmetric bubbles, and coexisting asymmetric attractors, to name a few. The control of multistability is also performed by using the so‐called linear augmentation scheme. Probe Simulation Program with Integrated Circuits Emphasis (Pspice) circuit simulations are carried out to verify the theoretical analyses. Furthermore, a chaos‐based image encryption is investigated using pseudorandom numbers generated by the proposed chaotic circuit and deoxyribonucleic acid (DNA) encoding technique.