Cubic-quintic Nonlinear Schrodinger Equation Research Articles

Applications of complete discrimination system approach to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation with optical soliton and bifurcation analysis

In this research, the complete discriminant system (CDS) of polynomial method (CDSPM) will be applied to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) with a particular emphasis on the introduction of various optical solitons and wave structures. Our analysis of the CQNLSE-AC illustrates the importance of CDSPM, and we give dynamic results, such as critical conditions and bifurcation points for solutions. Additionally, we determine several types of optical soliton solutions, including Jacobian elliptic function (JEF), hyperbolic function, and trigonometric function solutions, and also convert JEF into solitary wave (SW) solutions.

Dark pulse generation in mode-locked Erbium-doped fiber laser with Molybdenum Aluminium Boride coated side-polished fiber

We demonstrate the generation of dark pulses from an Erbium-doped fiber laser (EDFL) using a Molybdenum Aluminium Boride (MoAlB) as a saturable absorber for the first time. Molybdenum Aluminium Boride (MoAlB) was coated onto side-polished fiber (SPF) to improve the birefringence and nonlinearity inside the EDFL cavity with net anomalous dispersion. The soliton dark pulses are obtained from the laser cavity based on cubic-quintic nonlinear Schrödinger equation (CQNLSE) mechanism. The laser operated at 1560.6 nm with a pulse duration of 152 ns and repetition rate of 1.87 MHz as the pump power is set within 200 mW to 370 mW. At the maximum pump power, the average output power and pulse energy are recorded at 6.93 mW and 3.77 nJ, respectively. We think that this research is the first to show MoAlB 's potential for dark pulse generation.

Optical modulation in Er-doped fiber laser delivering dark pulse utilizing side polished fiber coated with Ti3AlC2 as saturable absorber

Dark pulse mode-locked Erbium doped fiber laser (EDFL) was demonstrated in this work by employing side polished fiber (SPF) coated with Ti3AlC2 solution using drop casting method. The modulation depth of fabricated SPF coated with Ti3AlC2 solution was characterized at 2.2%. This SPF coated structure allowed the formation of dark mode-locked pulse with improved the birefringence and nonlinearity in EDFL. The dark pulse generated was cubic-quintic nonlinear Schrödinger equation (CQNLSE) dark pulse with center wavelength at 1558.935 nm, repetition rate 1.839 MHz and pulse width 170.2 ns. Pulse stability was examined, revealing a high signal to noise ratio (SNR) of 65.86 dB. This work could serve as a foundation for the development of sustainable industry, innovation and infrastructure.

Rogue waves dynamics of cubic–quintic nonlinear Schrödinger equation with an external linear potential through a modified Noguchi electrical transmission network

In this paper, we have considered a modified Noguchi nonlinear transmission network with a dispersive element. Using the reductive perturbation method in the semi-discrete limit, we show how wave dynamics in the network can be governed by a cubic–quintic nonlinear Schrödinger equation (CQ-NLSE) with an external linear potential. Through the modified Darboux transformation, we constructed the first, second and third order rogue wave solutions. Although the linear potential function Γ(τ) does not influence the instability criterion of waves propagating in the network, it has a significant effect on the dynamics of the amplitude, phase and shirp of rogue waves. Our results also reveal that it is possible both to predict the position where the amplitude of rogue waves reaches its maximum value and to reduce their spatial width.

Stabilizing solitons of the cubic–quintic nonlinear Schrödinger equation by frequency-dependent linear gain-loss and delayed Raman response

We demonstrate transmission stabilization against radiation emission by frequency-dependent linear gain-loss and perturbation-induced frequency shifting for solitons of the cubic–quintic nonlinear Schrödinger (CQNLS) equation. We consider soliton propagation in a nonlinear optical waveguide with focusing cubic nonlinearity, defocusing quintic nonlinearity, and dissipative perturbations due to weak frequency-dependent linear gain-loss, cubic loss, and delayed Raman response. The frequency shifting is induced by delayed Raman response. Our perturbation analysis and numerical simulations with the perturbed CQNLS equation show that transmission stabilization with CQNLS solitons is indeed possible, and in this way provide the first demonstration of the stabilization method for solitons of a nonintegrable nonlinear wave model. Moreover, we find that transmission stabilization with energetic CQNLS solitons is realized with significantly smaller frequency shifts and pulse distortion compared with stabilization with energetic solitons of the cubic nonlinear Schrödinger equation. Therefore, our study also demonstrates that stabilization of energetic solitons by the method is significantly improved by the presence of defocusing quintic nonlinearity.

Multi-soliton interactions, comb behavior, strange mixed breathers and other exotic rogue waves generated in a nonlinear left-handed transmission line

We establish a cubic–quintic-nonlinear Schrödinger equation (CQNLSE) including cubic- and quintic-nonlinearities, second-, third- and fourth-order dispersions in a left-handed nonlinear transmission line. We use the collective coordinates’ theory and the Gaussian Ansatz function with six coordinates to improve the comprehension of the system. Besides, the following innovations have been found when linear and nonlinear effects interact. (i) Some phenomena have emerged such as comb behavior, fragmentation of waves and wall of waves. (ii) Some solutions have also emerged such as the dark soliton, multi-peak waves, Kuznetsov–Ma breathers, periodical and non periodical mixed wave trains, Sasa–Satsuma waves, Peregrine waves, and crossed solitons. Otherwise, the internal perturbations leading to all aforementioned exotic rogue events are well studied in detail.

DYNAMICAL EVOLUTION STUDY FOR ONE-DIMENSIONAL SYSTEM WITH CUBIC-QUINTIC NONLINEARITY

Nonlinear Schrodinger equation (NLSE) is a quantum system of nonlinear evolution， a kind of important equations, and in many areas have very important applications. Therefore, this type of equation models has important physical significance. People realized with GGPE describing BCS-BEC crossover, GGPE began to be utilized in people's research. When GPE and polytropic approximation are combined, the nonlinear term becomes generalized cubic-quintic nonlinearity. And the model becomes one-dimensional generalized cubic-quintic nonlinear Schrodinger equation (GCQ-NLSE). Many prior work deal with the special circumstances GCQ-NLSE the CQ-NLSE, and need to introduce additional integrability condition. In this work, we use the phase and amplitude coupling Transform combined with F-expansion method to solve the one-dimensional GCQ-NLSE. We obtained its soliton solutions without introducing any integrable condition. We apply the analytical solution obtained to our quantitative analysis of dynamic behavior of the model, whose results are dependent on the polytropic index which is related to the intensity of soliton strength, speed, system phonon velocity, these are depicted pictorially.

Modulational instability of truncated Airy pulses in cubic–quintic nonlinear optical waveguides with multiphoton absorptions

In this work we study the modulational instability (MI) phenomenon of truncated Airy pulses in a cubic–quintic nonlinear optical waveguide including nonlinear multiphoton absorptions as the two-photon absorption (TPA) and the three-photon absorption (3PA) in the anomalous group-velocity dispersion (AGVD) regime of chromatic dispersion. The investigation is done through a lossless and dimensionless cubic–quintic nonlinear Schrödinger equation (CQNLSE) where we launch a perturbed truncated Airy pulse. The perturbation is defined through the linear analysis of the steady-state of a continuous-wave (CW) in which we have chosen to operate with the optimal frequencies of the MI gain. The data obtained through this analysis led to generate the destabilization of the pulse through its lobes starting with the dominant peak. This MI is improved by the cooperating nonlinearities while its distance of achievement is reduced in this case growing from the main lobe to secondary sides lobes of tail. The MI is avoided in the competing case of nonlinearities for small peak power because the negative quintic Kerr nonlinearity (QKN) reduces or almost cancels the cubic Kerr nonlinearity (CKN) effect. If the peak power of the pulse increases, the deleterious effect of the competing QKN decreases while the AGVD and the CKN take over to bring the MI into existence breaking the Airy pulse into sub-pulses. This feature becomes more and more noised as the peak power increases. The TPA comes as a limiting factor of the MI for both the cases of cooperating and competing nonlinearities. The introduction of 3PA performs this TPA effect in the case of cooperating nonlinearities while in the competing case, the 3PA surprisingly cancels the TPA, increasing the AGVD effect. Since the same feature has already been observed recently in Ref. Heuteu et al, J. Opt. 23, 095503 (2021) for the supercontinuum generation (SCG) of Airy pulses, it is therefore shown that the 3PA could be used for better control the TPA effect even in the MI of this kind of pulses. Moreover, it is also shown that the increase of the Airy pulse duration drastically enhances its MI occurrence with noised outputs for large values of pulse width. However, reducing the pulse width also reduces its input energy, consequently the Airy pulse does not undergo the MI phenomenon but simply manifests the property of acceleration.

Chirped periodic waves for an cubic-quintic nonlinear Schrödinger equation with self steepening and higher order nonlinearities

In this paper, we study the cubic-quintic nonlinear Schrödinger equation (CQ-NLSE) to describe the propagation properties of nonlinear periodic waves (PW) in an optical fiber. We find chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF). We also obtain some solitary waves (SW) like dark, bright, hyperbolic and singular solitons. The chirp that corresponds to each of these optical solitons is also determined. The pair intensity is shown to be related to the nonlinear chirp, which is determined by self-frequency shift and pause self-steepening (SS). The shape of profile for these waves will also be display.

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.

Dromion−like structures in a cubic−quintic nonlinear Schrödinger equation using analytical methods

We investigate the dromion-like excitations corresponding to intramolecular chain-like proteins. In the present work, the dromion-like excitations are described by using cubic-quintic nonlinear Schrodinger equation (CQNSE) governing the dynamics of proteins and we analytically analyze the velocity (v) of dromion-like structure compared with velocity ( $$v_a$$ ) of acoustical sound waves corresponding to the longitudinal vibrations of protein molecules. Our work is motivated by the effectiveness and powerful mathematical techniques such as modified extended tanh function method and sine–cosine function method for solving CQNSE to obtain dromion-like structures.

Kink and Anti-Kink Soliton for Stationary Cubic-Quintic Discrete Nonlinear Schrodinger Equation

Kink and Anti-Kink Soliton for Stationary Cubic-Quintic Discrete Nonlinear Schrodinger Equation

Peregrine Solitons on a Periodic Background in the Vector Cubic-Quintic Nonlinear Schrödinger Equation

We present exact explicit Peregrine soliton solutions based on a periodic-wave background caused by the interference in the vector cubic-quintic nonlinear Schrodinger equation involving the self-steepening effect. It is shown that such periodic Peregrine soliton solutions can be expressed as a linear superposition of two fundamental Peregrine solitons of different continuous-wave backgrounds. Because of the self-steepening effect, some interesting Peregrine soliton dynamics such as ultrastrong amplitude enhancement and rogue wave coexistence are still present when they are built on a periodic background. We numerically confirm the stability of these analytical solutions against non-integrable perturbations, i.e., when the coefficient relation that enables the integrability of the vector model is slightly lifted. We also demonstrate the interaction of two Peregrine solitons on the same periodic background under some specific parameter conditions. We expect that these results may shed more light on our understanding of the realistic rogue wave behaviors occurring in either the fiber-optic telecommunication links or the crossing seas.

Rogue waves and modulation instability in an extended Manakov system

We obtain the general nth-order rogue wave solutions of the vector cubic-quintic nonlinear Schrodinger equation with self-steepening, alias extended Manakov system, by means of a nonrecursive Darboux transformation scheme. We show that in such a two-component system, owing to the presence of the self-steepening effect, there would emerge an anomalous Peregrine soliton state on one wave component whose peak can grow three times higher than its background level, at the expense of a heavy falling-off on the other wave component. We also demonstrate other interesting rogue wave dynamics such as coexisting, doublet, quartet, and sextet rogue waves, which depend on the choice of structural parameters. In addition, the modulation instability responsible for the formation of rogue waves is discussed, revealing the broad-range existence of rogue waves, irrespective of what dispersion situations considered.

Engineering rogue waves with quintic nonlinearity and nonlinear dispersion effects in a modified Nogochi nonlinear electric transmission network.

A one-dimensional modified Nogochi nonlinear electric transmission network with dispersive elements that consist of a large number of identical sections is theoretically studied in the present paper. The first-order nonautonomous rogue waves with quintic nonlinearity and nonlinear dispersion effects in this network are predicted and analyzed using the cubic-quintic nonlinear Schrödinger (CQ-NLS) equation with a cubic nonlinear derivative term. The results show that, in the semidiscrete limit, the voltage for the transmission network is described in some cases by the CQ-NLS equation with a derivative term that is derived employing the reductive perturbation technique. A one-parameter first-order rational solution of the derived CQ-NLS equation is presented and used to investigate analytically the dependency of the characteristics of the first-order rouge wave parameters on the electric transmission network under consideration. Our results show that when we change the quintic nonlinear and nonlinear dispersion parameter, the first-order nonautonomous rogue wave transforms into the bright-like soliton. Our results also reveal that the shape of the first-order nonautonomous rogue waves does not change when we tune the quintic nonlinear and nonlinear dispersion parameter, while the quintic nonlinear term and nonlinear dispersion effect affect the velocity of first-rogue waves and the evolution of their phase. We also show that the network parameters as well as the frequency of the carrier voltage signal can be used to manage the motion of the first-order nonautonomous rogue waves in the electrical network under consideration. Our results may help to control and manage rogue waves experimentally in electric networks.

Coupled cubic-quintic nonlinear Schrödinger equation: novel bright–dark rogue waves and dynamics

In this paper, we investigate the coupled cubic-quintic nonlinear Schrodinger equation, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. The breather wave solutions and novel bright–dark rogue wave solutions can be constructed by using the Darboux-dressing transformation and asymptotic expansion. These solutions show the spatiotemporal patterns of novel bright–dark rogue waves. It is demonstrated that the coupled or multi-component systems contain more interesting rogue wave phenomena than single component systems. Our results can be of importance in understanding and predicting the rogue waves of coupled cubic-quintic nonlinear Schrodinger equation.

Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations

We consider the Cauchy problem for the nonlinear Schrödinger (NLS) equation with double nonlinearities with opposite sign, with one term mass-critical and the other term mass-supercritical and energy-subcritical, which includes the well-known two-dimensional cubic-quintic NLS equation arising in the study of the boson gas with 2- and 3-body interactions. We prove global well-posedness and scattering in H1(Rd) below the threshold for nonradial data when 1≤d≤4.

Manipulating rogue waves, breathers and solitons in several non-integrable nonlinear Schrödinger equations

We study a non-integrable system in a resonant four-level media at room temperature. Under appropriate parameter conditions, the system can be reduced to the saturated nonlinear Schrodinger equation (SNLSE) and cubic-quintic nonlinear Schrodinger equation (CQNLSE). And we find a scheme to produce (1 + 1)-dimensional [(1 + 1) − D] spatial optical rogue waves, breathers, and solitons in these equations. The generation and disappearance of different nonlinear modes can be manipulated just by adjusting the amplitude and width of the initial Gaussian pulse. We also prove this scheme adapts to not only the integrable model, but also the non-integrable model. In addition, we also find the relation among these different nonlinear modes.

On stability properties of the Cubic-Quintic Schródinger equation with <inline-formula><tex-math id="M1">\begin{document}$\delta$\end{document}</tex-math></inline-formula>-point interaction

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrodinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index.

Exact Travelling Wave Solutions of Two Nonlinear Schrödinger Equations by Using Two Methods

The special kind of (G’/G)-expansion method and the new mapping method are easy and significant mathematical methods. In this paper, exact travelling wave solutions of the higher order dispersive Cubic-quintic nonlinear Schrodinger equation and the generalized nonlinear Schrodinger equation are studied by using the two methods. Finally, the solitary wave solutions, singular soliton solutions, bright and dark soliton solutions and periodic solutions of the two nonlinear Schrodinger equations are obtained. The results show that this method is effective for solving exact solutions of nonlinear partial differential equations.