Cubic-quintic Nonlinearity Research Articles

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.

Periodic Solutions of Duffing-Type Oscillators Using Continuous Piecewise Linearization Method

A new analytic algorithm called ‘continuous piecewise linearization method’ (CPLM) is developed to obtain periodic solutions of freely vibrating Duffing-type oscillators. This simple analytic algorithm is based on continuous piecewise linearization of the nonlinear stiffness with respect to displacement and was shown to produce very accurate results for few iterations. The algorithm is valid for Duffing-type oscillators possessing strong nonlinearity and/or undergoing large-amplitude oscillations. Studies conducted on Duffing oscillators with cubic, cubic-quintic and trigonometric sine stiffness nonlinearities showed that the CPLM results match standard numerical solutions and is more accurate than the popular energy balance method (EBM). Additionally, the present analysis shows that the CPLM is capable of predicting the quasi-linear behaviour observed in the oscillation history of Duffing-type oscillators with strongly nonlinearity and/or large-amplitude oscillations. This quasi-linear behaviour cannot be predicted by the EBM to which the CPLM is compared.

Generation of localized patterns in anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion via a variational approach.

We use the time-dependent variational method to examine the formation of localized patterns in dynamically unstable anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion. The governing equation is an extended nonlinear Schrödinger equation known for modified Frankel-Kontorova models of atomic lattices and here derived from an extended Bose-Hubbard model of bosonic lattices with local three-body interactions. In presence of modulated waves, we derive and investigate the ordinary differential equations for the time evolution of the amplitude and phase of dynamical perturbation. Through an effective potential, we find the modulationally unstable domains of the lattice and discuss the effect of the fourth-order dispersion in the dynamics. Direct numerical simulations are performed to support our analytical results, and a good agreement is found. Various types of localized patterns, including breathers and solitonic chirped-like pulses, form in the system as a result of interplay between the cubic-quintic nonlinearities and the second- and fourth-order dispersions.

Asymmetric wave transmission through one dimensional lattices with cubic-quintic nonlinearity

One dimensional lattice with an on-site cubic-quintic nonlinear response described by a cubic-quintic discrete nonlinear Schrödinger equation is tested for asymmetric wave propagation. The lattice is connected to linear side chains. Asymmetry is introduced by breaking the mirror symmetry of the lattice with respect to the center of the nonlinear region. Three cases corresponding to dimer, trimer and quadrimer are discussed with focus on the corresponding diode-like effect. Transmission coefficients are analytically calculated for left and right moving waves via backward transfer map. The different transmission coefficients for the left and right moving waves impinging the lattice give rise to a diode-like effect which is tested for different variations in asymmetry and site dependent coefficients. We show that there is a higher transmission for incoming waves with lower wavenumbers as compared to the waves with comparatively larger wavenumbers and a diode-like effect improves by increasing the nonlinear layers. We also show that in the context of transport through such lattices, the cooperation between cubic and quintic nonlinear response is not “additive”. Finally, we numerically analyse Gaussian wave packet dynamics impinging on the CQDNLS lattice for all three cases.

Families of fundamental and vortex solitons under competing cubic-quintic nonlinearity with complex potentials

We investigate the properties of two-dimensional(2D) fundamental and vortex solitons propagating in PT symmetric lattices with the competing cubic-quintic nonlinearity. We discuss the influence of the competing nonlinearity and the gain-loss coefficient on the existence and stability of both 2D fundamental solitons and vortex solitons, and obtain the existence and stability ranges of them. The stability of solitons under different competing CQ nonlinearity is analyzed by using the VK criterion or the anti-VK criterion. We demonstrate that the whole nonlinearity is determined by the coefficients of nonlinear terms and the propagation constants of solitons. In particular, the power curve of vortex soliton leap and move back near the Bloch band instead of bifurcating from it under the defocusing cubic and focusing quintic nonlinearity.

Spectral dynamics of square pulses in passively mode-locked fiber lasers

We investigate experimentally and numerically the spectral dynamics of square pulses generated in passively mode-locked fiber lasers under the dissipative soliton resonance. The features of the transition from the single-peak spectral profile to the doublet spectrum with increasing pump power are studied. The used master equation takes into account the gain saturation, the quadratic frequency dispersion of the gain and the refractive index, and the cubic-quintic nonlinearity of the losses and refractive index. Experimental data are obtained for an Er:Yb-doped fiber ring laser. The theoretical and experimental results are in good agreement with each other.

Multi-pole dark solitons in nonlocal and cubic-quintic nonlinear medium

In this paper, we mainly simulate the characteristics of the ground state dark soliton and the multipole dark soliton in the nonlocal and cubic-quintic nonlinear medium. Firstly, the influences of the degree of nonlocality on the amplitude and the width of the dark soliton in the self-defocusing cubic-and self-focusing quantic-nonlinear medium are studied. Secondly, we find the nonlinear parameters affecting the amplitude values of solitons, but the refractive index induced by the light beam is always a fixed value. The numerical results show that the ground state dark soliton can be propagated stably alone the z axis, and the stable states of the dipole soliton and the dark tri-pole and quadru-pole solitons are stable. However, some quadru-pole dark soliton is unstable after propagating the remote distance. Furthermore, we also discuss the characteristics of the ground state dark soliton and the dark dipole soliton in the local cubic-nonlinear and nonlocal quantic nonlinear media. Both the amplitude and the beam width of the dark ground state soliton and dark dipole soliton are also affected by the degree of nonlocality and nonlinearity. Two boundary values of the induced refractive index change with the variations of the three nonlinear parameters. The dark soliton and the dipole dark soliton are more stable in the self-focusing cubic nonlinear and the nonlocal self-defocusing quantic nonlinear medium than those in the self defocusing cubic nonlinear and nonlocal self-focusing quantic nonlinear medium. The powers of single dark soliton and dark tri-pole soliton decrease monotonically with the increase of propagation constant when the cubic-quintic nonlinearities are certain values and these degrees of nonlocalities are taken different values. Furthermore, we also analyze linear stabilities of various nonlocal spatial dark solitons. And we find that the dipole dark soliton is unstable when the propagation constant is in the region[-0.9,-1.0]. These properties of linear stabilities of other multi-pole dark solitons are the same as their propagation properties.

He’s homotopy perturbation method for solving time fractional Swift-Hohenberg equations

This paper find the most effective method to solve the time-fractional Swift-Hohenberg equation with cubicquintic non-linearity by combining the homotopy perturbation method and the fractional complex transform. The solution reveals some intermittent properties of thermal physics.

Analytic methods to find beating transitions of asymmetric Gaussian beams in GNLS equations.

In a simple model of propagation of asymmetric Gaussian beams in nonlinear waveguides, described by a reduction to ordinary differential equations of generalized nonlinear Schrödinger equations with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile, the beam widths exhibit two different types of beating behavior, with transitions between them. We present an analytic model to explain these phenomena, which originate in a 1:1 resonance in a 2 degree-of-freedom Hamiltonian system. We show how small oscillations near a fixed point close to 1:1 resonance in such a system can be approximated using an integrable Hamiltonian and, ultimately, a single first order differential equation. In particular, the beating transitions can be located from coincidences of roots of a pair of quadratic equations, with coefficients determined (in a highly complex manner) by the internal parameters and initial conditions of the original system. The results of the analytic model agree with the numerics of the original system over large parameter ranges, and allow new predictions that can be verified directly. In the CQ case, we identify a band of beam energies for which there is only a single beating transition (as opposed to 0 or 2) as the eccentricity is increased. In the SAT case, we explain the sudden (dis)appearance of beating transitions for certain values of the other parameters as the grade-index is changed.

Strongly localized dark modes in binary discrete media with cubic-quintic nonlinearity within the anti-continuum limit

The existence of dark strongly localized modes of binary discrete media with cubic-quintic nonlinearity is numerically demonstrated by solving the relevant discrete nonlinear Schrödinger equations. In the model, the coupling coefficients between adjacent sites are set to be relatively small representing the anti-continuum limit. In addition, approximated analytical solutions for vectorial solitons with various topologies are derived. Stability analysis of the localized states was performed using the standard linearized eigenfrequency problem. The prediction from the stability analysis are furthermore verified by direct numerical integrations.

Jacobi elliptic solutions, solitons and other solutions for the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity

Based on several integration tools, namely the Riccati equation method, the Bernoulli equation method, the extended auxiliary equation method, the new mapping method and the $\phi^{6}$ -model expansion method, we obtain many exact solutions including the optical bright-dark-singular soliton solutions, Jacobi elliptic solutions and trigonometric function solutions of the nonlinear Schrodinger equation (NLSE) with fourth-order dispersion and cubic-quintic nonlinearity, self-steeping and self-frequency shift effects which describes the propagation of an optical pulse in optical fibers. We compare the results yielding from these integration tools together with each others. Also, a comparison between our results in this paper and the well-known results are given.

Theory of passively-mode-locked fiber lasers with phase-modulated square pulses

We present results of the analysis of square pulses generated in passively-mode-locked lasers under dissipative-soliton-resonance conditions. The master equation used in this work takes into account the gain saturation, the quadratic frequency dispersion of the gain and the refractive index, and the cubic-quintic nonlinearity of the losses and refractive index. The phase modulation effects are defined as a perturbation to the solution of this equation that has the form of a square pulse without phase modulation. An analytical dependence of the change in the carrier frequency of the radiation along the pulse is found. Conditions for the occurrence of the singlet and doublet spectra of the square pulse are determined. The mechanisms responsible for the occurrence of this regime are analyzed. The obtained analytical dependencies are in good agreement with the results of numerical simulation.

Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability

The higher-order nonlinear Schrodinger equation (NLSE) with fourth-order dispersion, cubic-quintic terms, self-steepening and nonlinear dispersive terms describes the propagation of extremely short pulses in optical fibers. In this paper, the elliptic function, bright and dark solitons and solitary wave solutions of higher-order NLSE are constructed by employing a modified extended direct algebraic method, which has important applications in applied mathematics and physics. Furthermore, we also present the formation conditions of the bright and dark solitons for this equation. The modulation instability is utilized to discuss the stability of these solutions, which shows that all solutions are exact and stable. Many other higher-order nonlinear evolution equations arising in applied sciences can also be solved by this powerful, effective and reliable method.

Dual-switching behavior of nonlocal interfaces

Nonlinear interfaces separating two diffusive Kerr-type media exhibit dual switching between total internal reflection and transmission. This property is found within a weakly nonlocal regime when both a nonparaxial treatment of the problem and a full two-dimensional model for carrier diffusion are assumed. The theoretical model is shown to predict an effective cubic-quintic nonlinearity with competing terms that produces such property. The validity of the analysis is contrasted with a full set of numerical simulations.

Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion

We study the generation of dissipative solitons (DSs) in the model of the fiber-laser cavities under the combined action of cubic-quintic nonlinearity, multiphoton absorption and/or multiphoton emission (nonlinear gain), and gain dispersion. A random component of the group-velocity dispersion (GVD) is included too. The DS creation and propagation are studied by means of a variational approximation and direct simulations, which are found to be in reasonable agreement. With a proper choice of the gain, robust DS operation regimes are predicted for different combinations of multiphoton absorption and emission, in spite of the presence of the perturbation in the form of the random GVD. Importantly, the zero background around the solitons remains stable in the presence of the (necessary) linear gain. The solitons are stable too against a certain (realistic) level of noise. Another essential finding is that the quintic gain in the form of three-photon emission (3PE) offers an alternative mechanism for supporting stable solitons, provided that it is not too strong. The DSs coexist in low- and high-amplitude forms, for a given value of their width. The low-amplitude DS is stable, while its high-amplitude counterpart is subject to the blowup instability, in the presence of the 3PE. Interactions between DSs show various scenarios of the creation of breather states through merger of the two solitons.

Anti-continuum approach on odd solitons in binary discrete media with cubic-quintic nonlinearity

The existence and stability of bright odd strongly localized modes in discrete binary media with cubic and quintic nonlinearities are investigated mainly by applying anti-continuum method. There are three different types of bright localized modes being studied; symmetric, antisymmetric and shifted odd modes. However, certain analytical analysis is carried out first applying the anti-continuum limit assumption to estimate initial excitations in the media. It can be deduced that, the quintic nonlinearity is a significant factor for stability in antisymmetric and shifted odd modes.

Periodic waves and their stability in competing cubic-quintic nonlinearity

Exact expressions are obtained for stationary periodic waves in cubic-quintic nonlinear media with opposite signs of nonlinearity. It is shown that defocussing/focussing quintic nonlinearity can lead to stabilization of snoidal/(cnoidal and dnoidal) periodic waves depending on whether the Kerr medium is focusing/defocusing. Direct numerical simulations confirm results of the linear stability analysis. Different stable patterns e.g. bright solitons and kinks are also obtained under certain conditions.

Exact solutions and optical soliton solutions for the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity

In this article, we apply the soliton ansatz method combined with the Jacobi elliptic equation method which is different from the F-expansion method to obtain several types of Jacobi elliptic function solutions, the optical bright-dark-singular soliton solutions and trigonometric function solutions of the nonlinear Schrodinger equation with fourth-order dispersion and cubic-quintic nonlinearity, self-steeping and self-frequency shift effects which describes the propagation of an optical pulse in optical fibers. Comparison between our results in this article and the well-known results are given.

Stability, integrability, and nonlinear dynamics of PT-symmetric optical couplers with cubic cross-interactions or cubic-quintic nonlinearities.

We explore the parity-time-( PT)-symmetric optical couplers with the cubic both self- and cross-interactions corresponding to self- and cross-phase modulations. When the coefficient of the cubic cross-interaction is chosen as the different values, we find three distinct cases for two branches, including the stable-stable modes (linear unbroken PT-symmetric phase), stable-unstable modes (linear unbroken PT-symmetric phase), as well as unstable-unstable modes (linear broken PT-symmetric phase). Moreover, we find the periodic trajectories for some parameters. Similarly, we also explore the PT-symmetric optical couplers with cubic-quintic self-phase modulations. We numerically give the stable and unstable regions of the cubic-quintic system. Moreover, we also find the periodic trajectories for some parameters in the Stokes domain.

Bouncing dynamics of Bose–Einstein condensates under the effects of gravity

Bouncing dynamics of quasi-one dimensional Bose–Einstein condensates (BECs) falling under gravity on delta-function potentials is investigated. First, we consider a single component BEC in the presence of cubic-quintic nonlinearity and study dynamical behavior of different parameters of the system using variational and numerical approaches. We see that the quintic nonlinearity plays a dominant role over cubic nonlinear interaction to extend the bouncing dynamics in the non-linear regime. We find that a matter-wave performs bouncing motion only for certain discrete values of initial position above the reflecting potential. We then consider bouncing dynamics of binary BECs. It is shown that the pair of matter-waves bounces together if inter-species interaction is attractive. However, their pairing breaks down if the inter-species interaction is made repulsive.