Quadratic-cubic Nonlinearity Research Articles

Exact solutions to the resonant nonlinear Schrödinger equation with time-dependent coefficients under dual-power law nonlinearity

Abstract In this paper, the resonant nonlinear Schrödinger equation with time-dependent coefficients which appears in the research process of Madelung fluid is probed by means of two different methods. As an outcome, dozens of distinct types of complex exact solutions are built in the condition of taking dual-power law nonlinearity into account. More specifically, a range of complex solitary, soliton as well as elliptic wave solutions are offered in terms of the unified method, and a series of hyperbolic, triangular, Jacobi elliptic doubly periodic, rational as well as exponential type solutions are given in the light of the improved modified extended tanh-function method. Moreover, the dual-power law nonlinearity, it should be noted, can degenerate to other popular nonlinearity forms including the kerr, power, parabolic law nonlinearities and quadratic-cubic nonlinearity with specific values of concerned parameters, and we also consider these peculiar circumstances in the solving process. Finally, we draw 2D, 3D and contour images for distinct types of solutions that we acquired to simplify the physical interpretation.

Nonlinear tunneling of chirped similaritons in non-centrosymmetric waveguides with quadratic-cubic nonlinearity

The nonlinear tunneling of the optical similaritons through both dispersion and nonlinear barriers in a non-centrosymmetric waveguide exhibiting second- and third-order nonlinearity is studied. Within the framework of an inhomogeneous quadratic-cubic nonlinear Schrödinger equation with distributed group velocity dispersion, reciprocal of the group velocity, quadratic-cubic nonlinearity, and gain or loss, we demonstrate the existence of a rich variety of exact similariton solutions in the shape of bright, kink, anti-kink, W-shaped, and gray solutions. The results show that these self-similar waves involve certain control parameters in their amplitude, phase, width, and shift of the inverse group velocity, which enables us to control their evolution dynamics in the waveguide system through a suitable choice of these parameters. It is found that the parameter of reciprocal of the group velocity decides the phase shift and group velocity of these self-similar waves. Also, the stability of the solutions is discussed numerically. Results of this paper are helpful for enriching the similariton theory and understanding the dynamics of self-similar waves in systems with quadratic–cubic nonlinearities.

Method of searching for a W-shaped like soliton combined with other families of solitons in coupled equations: application to magneto-optic waveguides with quadratic-cubic nonlinearity

Method of searching for a W-shaped like soliton combined with other families of solitons in coupled equations: application to magneto-optic waveguides with quadratic-cubic nonlinearity

Optical solitons of the (2+1)-dimensional nonlinear Schrödinger equation with spatio-temporal dispersion in quadratic-cubic media

The major objective of this paper is to study the perturbed nonlinear Schrödinger equation with spatio-temporal dispersion in (2+1)-dimensional by using the complete discrimination system for polynomial. This paper discusses the propagation patterns in nonlinear fibers with quadratic-cubic nonlinearity. Additionally, a host of optical wave solutions are obtained, where solitary wave patterns and elliptic functions double periodic patterns are newly received. Besides, some two dimensional figures of solutions are illustrated in parameter space. All these solutions and figures are available to investigate the propagation dynamics of optical solitons in optical fibers.

Discussion on localized pulses for third-order nonlinear Schrödinger equation and negative index material with quadratic cubic nonlinearity

In this paper, we study the third-order nonlinear Schrödinger equation (NLSE) and optical metamaterial (OM) with quadratic-cubic nonlinearity (OM-QCNL) to describe the properties and presence of propagating solitary waves (SW) in a fibre optic medium. We present an effective field amplitude ansatz that allows us to obtain SW using the governing model. More specifically, in the presence of all orders of dispersion and various nonlinear variables, we find SW solutions of dipole, bright, dark and other forms. The graphical representation of these waves will also be presented.

Diverse optical solitons to nonlinear perturbed Schrödinger equation with quadratic-cubic nonlinearity via two efficient approaches

In this study, the nonlinear perturbed Schrödinger equation(NPSE) with nonlinear terms as quadratic-cubic law nonlinearity media with the beta derivative is investigated and this investigated model is considered an icon in the field of optical fibers where it describes the wave function or state function of the optical system. Numerous solutions are extracted by engaging two novel schemes known as generalized -expansion method (GEEM) and rational extended sinh-Gordon equation expansion method (REShGEEM) in distinct forms such as bright, dark, singular and combinations of these solutions. In addition, plane wave, periodic and exponential solutions are also recovered. Using the computer application Wolfram Mathematica 11, the dynamic structure and physical characterization of some observed solutions are vividly depicted by sketching different plots. Comparing our new results with well-known literature is also given which justifies the novelty of this work. Obtained results will hold a significant place in the field of nonlinear optical fibers and suggest that the proposed methods are very influential and effective tools for solving more complex nonlinear partial differential equations in mathematical physics, engineering and nonlinear optics.

Highly Dispersive Optical Solitons in Fiber Bragg Gratings with Quadratic-Cubic Nonlinearity

Highly dispersive solitons in fiber Bragg gratings with quadratic-cubic law of nonlinear refractive index are studied in this paper. The G′/G-expansion approach and the enhanced Kudryashov’s scheme have made this retrieval possible. A deluge of solitons, that emerge from the two integration schemes, are presented.

Coupled sn-cn, sn-dn, cn-dn Jacobi elliptic functions and solitons solutions in magneto-optic waveguides with quadratic-cubic nonlinearity

The current paper is dedicated to the study of the propagation of progressive solitons in magneto-optical waveguides that carry the quartic-cubic nonlinearity described by a coupled system of of nonlinear Schrödinger equations. In this direction, we take the help of a modified F-expansion method which allows us to widen the class of solutions to these coupled equations and permit the propagation of different waves in the two channels. First of all, we constructed the solutions in terms of Jacobi elliptic functions in the forms of and After that, we obtained the progressive soliton solutions when these functions approached their limiting values with regards to the modulus of ellipticity. The progressive solutions include bright-dark soliton, bright-bright soliton, dark-dark soliton, bright-front soliton, dark-front soliton. In addition, we present the figures 1–6 which graphically exhibit the representative structures of each explicit solution for some special parameter values.

Transformation and interactions among solitons in metamaterials with quadratic-cubic nonlinearity and inter-model dispersion

In this paper, we investigate multiple soliton interactions and other solitary wave solutions (SWS) for a perturbed nonlinear Schrödinger equation (NLSE) with negative index material having quadratic-cubic nonlinearity (NLSE-QCN). Due to its high order dispersion term, this model yields sub-picosecond impulses useful in mode-locked ring lasers. Hirota bilinear method (HBM) will be used to study soliton interaction. By controlling the parameters, we will obtain [Formula: see text], [Formula: see text], parabolic and anti-parabolic, butterfly, bright and dark shaped solitons. On the other hand, we will obtain some other solitary wave solutions with the help of Sine-Gordon expansion (SGE) scheme.

Solitary and periodic waves in quadratic-cubic non-centrosymmetric waveguides

We present a wide class of solitary and periodic waves in a non-centrosymmetric waveguide exhibiting second- and third-order nonlinearities. We show the existence of bright, gray, and W-shaped solitary waves as well as periodic waves for the quadratic-cubic nonlinear Schrödinger equation. We also obtained the exact analytical algebraic-type solitary waves of this model, including bright and W-shaped waves. The results illustrate the propagation of potentially rich set of nonlinear structures through the optical system. Such privileged nonlinear waves characteristically exist due to a balance among diffraction, quadratic and cubic nonlinearities. The stability of the solutions is discussed numerically under finite initial perturbations. These results can be used in studying the propagation of shape-preserved and periodic waves in optical waveguides with quadratic-cubic nonlinearities.

Homoclinic breaters, mulitwave, periodic cross-kink and periodic cross-rational solutions for improved perturbed nonlinear Schrödinger's with quadratic-cubic nonlinearity

In this work, we study the soliton solutions for the improved perturbed nonlinear Schrödinger's equation (PNLSE) with quadratic-cubic law nonlinearity (QCLN) by utilizing the log transformation and symbolic computation with the ansatz function schemes. We obtain M -shaped rational soliton, double exponential form, periodic cross-kink (PCK), periodic cross-rational (PCR), multi-waves and homoclinic breather approach. We'll show that M -shaped rational solitons with kink and periodic waves have unique structures and highly interesting interactional behavior. Furthermore, we'll discuss the dynamics of obtained solutions with the help of graphs by assigning appropriate values to the parameters.

Propagating chirped lambert W-kink solitons for ac-driven higher-order nonlinear Schrödinger equation with quadratic-cubic nonlinearity

Propagating chirped lambert W-kink solitons for ac-driven higher-order nonlinear Schrödinger equation with quadratic-cubic nonlinearity

Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms

In this paper, a compelling pseudo-spectral approach namely split-step Fourier transform (SSFT) is utilized for the first time to report the complex Ginzburg Landau (CGL) equation. In this regard, six nonlinear forms of this notable equation are associated with this model, which are the kerr law nonlinearity, power law nonlinearity, quadratic-cubic nonlinearity, cubic-quintic nonlinearity, dual power nonlinearity, and polynomial law nonlinearity. Since opting for an analytical solution is complicated and only provided for a limited set of soliton solutions, the numerical solution might render a more generalized aspect in addressing versatile solutions for a wide range of arbitrary input pulses. Therefore, the SSFT scheme is efficiently employed to extract a plethora of optical solitons solutions such as kink waves, bright, dark, bright-dark, singular, singular periodic solitons. The obtained results, via MATLAB, are in total agreement with the findings explored form the exact analytical solutions. As a result, this numerical technique may provide a creditable mathematical tool to solve a wide range of other nonlinear evolution partial differential equations in the mathematical physics field.

Quadratic fractional solitons

We introduce a system combining the quadratic self-attractive or composite quadratic-cubic nonlinearity, acting in the combination with the fractional diffraction, which is characterized by its Lévy index α. The model applies to a gas of quantum particles moving by Lévy flights, with the quadratic term representing the Lee-Huang-Yang correction to the mean-field interactions. A family of fundamental solitons is constructed in a numerical form, while the dependence of its norm on the chemical potential characteristic is obtained in an exact analytical form. The family of quasi-Townes solitons, appearing in the limit case of α=1/2, is investigated by means of a variational approximation. A nonlinear lattice, represented by spatially periodical modulation of the quadratic term, is briefly addressed too. The consideration of the interplay of competing quadratic (attractive) and cubic (repulsive) terms with a lattice potential reveals families of single-, double-, and triple-peak gap solitons (GSs) in two finite bandgaps. The competing nonlinearity gives rise to alternating regions of stability and instability of the GS, the stability intervals shrinking with the increase of the number of peaks in the GS.

Construction of new exact solutions of the resonant fractional NLS equation with the extended Fan sub-equation method

The fucus of this study is to find a set of some novel solutions concerning the resonant fractional nonlinear Schrödinger equation (R-FNLSE) with quadratic-cubic nonlinearity by employing the extended FAN sub-equation approach. The parameter α∈(0,1] is the core constraint which simulate the flow rate propagation, plays a key rote in telecommunications and the theory of optical fibres. This equation expresses the gesture of solitons and Madelung fluids in various nonlinear systems. These outcomes are optical, bright, dark, explicit, periodic and combined wave solutions and efficiently demonstrated with the aid of 3D plots. The representation of these solutions is carried out in order to have an idea about the structure of such model and can also be extended to many other complex models of recent era.

Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

Linear and energy stable schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity based on a modified scalar auxiliary variable approach

Linear and energy stable schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity based on a modified scalar auxiliary variable approach

Optical solitons in birefringent fibers with quadratic-cubic nonlinearity by traveling waves and Adomian decomposition

This paper obtains the bright, singular and combo solitons solutions in birefringent fibers, with four-wave mixing, that maintains quadratic-cubic law of refractive of refractive index, by implementing traveling wave hypothesis. Adomian decomposition method is subsequently applied to the obtained results on bright solitons to display its numerical simulation with an impressive error measure.

Optical solitons in birefringent fibers with quadratic-cubic nonlinearity using three integration architectures

In this work, the nonlinear Schrödinger’s equation is studied for birefringent fibers incorporating four-wave mixing. The improved tanϕ(ξ)2-expansion, first integral, and G′G2-expansion methods are used to extract a novel class of optical solitons in the quadratic-cubic nonlinear medium. The extracted solutions are dark, periodic, singular, and dark-singular, along with other soliton solutions. These solutions are listed with their respective existence criteria. The recommended computational methods here are uncomplicated, outspoken, and consistent and minimize the computational work size, which give it a wide range of applicability. A detailed comparison with the results that already exist is also presented.

Hexagon Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift--Hohenberg Equation

Stationary fronts connecting the trivial state and a cellular (distorted) hexagonal pattern in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity are known to undergo a process of infinitely many folds as a parameter is varied, known as homoclinic snaking, where new hexagon cells are added to the core, leading to a region of infinitely-many, co-existing localized states. Outside the homoclinic snaking region, the hexagon fronts can invade the trivial state in a bursting fashion. In this paper, we use a far-field core decomposition to set up a numerical path-following routine to trace out the bifurcation diagrams of hexagon fronts for the two main orientations of cellular hexagon pattern with respect to the interface in the bistable region. We find for one orientation that the hexagon fronts can destabilize as the distorted hexagons are stretched in the transverse direction leading to defects occurring in the deposited cellular pattern. We then plot diagrams showing when the selected fronts for the two main orientations, aligned perpendicular to each other, are compatible leading to a hexagon wavenumber selection prediction for hexagon patches on the plane. Finally, we verify the compatibility criterion for hexagon patches in the Swift-Hohenberg equation with a quadratic-cubic nonlinearity and a large non-variational perturbation. The numerical algorithms presented in this paper can be adapted to general reaction-diffusion systems.