Cubic-quintic Nonlinear Schrodinger Research Articles

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.

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.

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.

Reduction to modified KdV and its KP-like generalization via phase modulation

The main observation of this paper is that the modified Korteweg–de Vries equation has its natural origin in phase modulation of a basic state such as a periodic travelling wave, or more generally, a family of relative equilibria. Extension to 2 + 1 suggests that a modified Kadomtsev–Petviashvili (or a Konopelchenko–Dubrovsky) equation should emerge, but our result shows that there is an additional term which has gone heretofore unnoticed. Thus, through the novel application of phase modulation a new equation appears as the 2 + 1 extension to a previously known one. To demonstrate the theory it is applied to the cubic-quintic nonlinear Schrödinger (CQNLS) equation, showing that there are relevant parameter values where a modified KP equation bifurcates from periodic travelling wave solutions of the 2 + 1 CQNLS equation.

The analytic solutions of Schrödinger equation with Cubic-Quintic nonlinearities

Abstract This paper deals with the nonlinear Schrodinger equation with Cubic-Quintic nonlinearities. After transforming the non-autonomous nonlinear Schrodinger equation into a stationary equation by adopting the BPVK idea, we give the classification of the solutions to the Cubic-Quintic nonlinear Schrodinger equation by utilizing the complete discrimination system for polynomial method. As an auxiliary result, we use the homotopy renormalization method (HTR) to obtain the global solutions to the Ermakov-Pinney equation satisfied by the width function.

Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method

The (2+1)-dimensional hyperbolic and cubic-quintic nonlinear Schrodinger equations describe the propagation of ultra-short pulses in optical fibers of nonlinear media. By using an extended sinh-Gordon equation expansion method, some new complex hyperbolic and trigonometric functions prototype solutions for two nonlinear Schrodinger equations were derived. The acquired new complex hyperbolic and trigonometric solutions are expressed by dark, bright, combined dark-bright, singular and combined singular solitons. The obtained results are more compatible than those of other applied methods. The extended sinh-Gordon equation expansion method is a more powerful and robust mathematical tool for generating new optical solitary wave solutions for many other nonlinear evolution equations arising in the propagation of optical pulses.

Solitons generated by self-organization in bismuth germanium oxide single crystals during the interaction with laser beam

We present here the experimental, theoretical, and numerical investigations of Kerr solitons generated by self-organization in black and yellow high quality bismuth germanium oxide (Bi12GeO20) single crystals. A picosecond laser beam of increasing power induces competing cubic and quintic nonlinearities. The numerical evolution of two-dimensional complex cubic-quintic nonlinear Schrodinger equation with measured values of nonlinearities shows the compensation of diffraction by competing cubic and quintic nonlinearities of opposite sign, i.e., the self-generation and stable propagation of solitons. Experiments as well as numerical simulations show higher nonlinearity in the black Bi12GeO20 than in the more transparent yellow one.

On optical solitons: the nonlinear Schrödinger equation with self-steepening and self-frequency shift

This article studies the dynamics of optical solitons in cubic-quintic nonlinear Schrodinger model that describes the propagation of femtosecond light pulses in optical fibers. The model is considered with self-steepening and self-frequency shift. As a result, the dark and singular optical solitons are extracted using \((G^{\prime }/G)\)-expansion scheme. Additionally, the constraint conditions for the existence of these solutions are also listed.

Flat top solitons on linear gaussian potential

The study of Nonlinear Schrodinger Equation has been wide focus from many researchers especially analysing the result of collision as it describes the soliton propagation. This paper considers the soliton scattering of cubic-quintic Nonlinear Schrodinger Equation on localized Gaussian potential. By applying Super-Gaussian ansatz as the trial function for variational approximation (VA) method, the soliton interaction may acquire flat-top shape with appropriate parameters. The result of VA will be compared to numerical analysis to check the accuracy of analytical predictions.

Bright solitons for the coupled cubic-quintic non-linear Schrödinger equations

In this paper, we investigate the coupled cubic-quintic non-linear Schrodinger equations, which describe the effects of quintic non-linearity on the ultra-short optical-pulse propagation in the non-Kerr medium. Bi-linear forms, bright one- and two-soliton solutions are obtained with Hirota method and symbolic computation. Via the asymptotic analysis, we find that the bright two solitons keep their original amplitudes, widths and velocities invariant after each collision except for the phase shifts. By virtue of the graphical analysis, head-on and overtaking collisions between the bright two solitons are presented. Bound-state soliton can also be observed with the breather-like structures.

Solitons in Dual-Barrier Magnetic Tunnel Junction

We have investigated the magnetic spin dynamics in a dual-barrier magnetic tunnel junction (MTJ), which consists of a free layer sandwiched between the top and bottom pinned layers. Current-induced spin transfer torque is produced when spin-polarised electrons flow through this spin valve pillar. We have formulated an extended Landau-Lifshitz-Gilbert-Slonczewski equation for dual-barrier spin valve multilayer structure. We have projected the unit sphere of spin onto a complex plane and transformed the equation of motion to a cubic-quintic nonlinear Schrodinger equation. The soliton solutions are constructed by employing Hirota’s Bilinearisation technique. The investigation shows that the amplitude of solitons depends greatly on the system parameters. Solitons emerge from this inhomogeneous nonlinear system representing a balance between the effect of dispersion and nonlinearity. The presence of these solitons as magnetic excitations in a ferromagnetic multilayer system may lead to noise-free nonvolatile data storage, data transmission and also in quantum computations.