Cubic-quintic Nonlinearity Research Articles

Orbital stability of solitary wave for the Zakharov equation with cubic-quintic nonlinearity

The orbital stability of solitary waves for the Zakharov equation with cubic-quintic nonlinearity is studied in this paper. Based on the theory of Grillakis–Shatah–Strauss on the orbital stability of solitary wave of nonlinear Hamiltonian systems, through detailed spectral analysis and technical computation, we overcome the difficulties caused by the quintic nonlinearity in the studied equation, and prove the conditions and assumption that the orbital stability of solitary wave solutions of the studied equations needs to satisfy. Further, we obtain the conclusion that the solitary waves of the Zakharov equation with cubic-quintic nonlinearity are orbitally stable under both parameter conditions.

Modulational instability for a cubic-quintic model of coupled Gross–Pitaevskii equations with residual nonlinearities

Abstract We demonstrate the existence of modulational instability (MI) in both trapped miscible and immiscible two component Bose-Einstein condensates. The study is addressed theoretically and numerically in the framework of one-dimensional coupled Gross-Pitaevskii equations incorporating intra- and interspecies cubic-quintic nonlinearities with higher order ones. Using the time-dependent variational approach, we derive the new Euler-Langrange equations for the time evolution of the phase and amplitude of the modulational perturbation as well as the effective potential and the instability criteria of the system. We examine the effects of higher-order nonlinearities on the instability dynamics of the condensates. We show that the modulational properties of the chosen wave numbers are significantly modified. Direct numerical simulations run by the split step Fourier method confirm the analytical predictions.

Dark gap solitons in bichromatic optical superlattices under cubic-quintic nonlinearities.

We demonstrate the existence of two types of dark gap solitary waves-the dark gap solitons and the dark gap soliton clusters-in Bose-Einstein condensates trapped in a bichromatic optical superlattice with cubic-quintic nonlinearities. The background of these dark soliton families is different from the one in a common monochromatic linear lattice; namely, the background in our model is composed of two types of Gaussian-like pulses, whereas in the monochromatic linear lattice, it is composed of only one type of Gaussian-like pulses. Such a special background of dark soliton families is convenient for the manipulation of solitons by the parameters of bichromatic and chemical potentials. The dark soliton families in the first, second, and third bandgap in our model are studied. Their stability is assessed by the linear-stability analysis, and stable as well as unstable propagation of these gap solitons are displayed. The profiles, stability, and perturbed evolution of both types of dark soliton families are distinctly presented in this work.

Exploration on modulation instability in 𝒫𝒯-symmetric non-Kerr Bragg grating structures

In this paper, we study Modulation Instability (MI) and its correlated gain spectrum with Parity-Time ([Formula: see text])-symmetry. The cubic-quintic nonlinearities in the Bragg grating structure produce ultrashort pulses. The MI evaluation started with a linear stability approach, and we investigated the effects of various system functions regarding the instability gain spectra at the top and bottom sides of the Photonic BandGap (PBG), in addition to the anomalous and normal dispersion domains. The interaction of cubic and quintic (focusing and de-focusing) nonlinearity is highlighted, resulting in a new type of MI band. Due to the consequences of quintic nonlinearity, an additional MI band appears for the three dynamics associated with the gain/loss function for the edges of the bottom PBG. After enforcing the [Formula: see text]-symmetry for the anomalous dispersion system, the dent transformed into a split under the influence of the power in the de-focusing quintic effect. Overall, MI gain and bandwidths across the entire MI gain spectrum are usually increasing. As an outcome, we are confident that these studies will aid in the generation of gap solitons in [Formula: see text]-symmetry regimes.

Bright soliton dynamics for the resonant nonlinear Schrödinger equation with generalized cubic-quintic nonlinearity

Bright soliton dynamics for the resonant nonlinear Schrödinger equation with generalized cubic-quintic nonlinearity

Schrödinger-Hirota equation in birefringent fibers with cubic-quantic nonlinearity and multiplicative white noise in the ito sense: Nucci’s reductions and soliton solutions

This paper explores innovative solutions for the Stochastic Schrödinger-Hirota equation within the context of birefringent fibers with cubic-quintic nonlinearity, emphasizing incorporating multiplicative white noise in the Itô sense. Leveraging the Nucci reduction method, the study focuses on obtaining exact solutions, shedding light on the intricate interplay between quantum mechanics and stochastic processes. The Nucci reduction method is a powerful tool to facilitate the derivation of precise solutions, showcasing its efficacy in unravelling complex mathematical structures and providing valuable insights into the behaviour of quantum systems under the influence of diverse parameters. In addition, two effective and convenient procedures are employed to extract bright, dark, and unique soliton solutions, as well as their combination. Exploring these solutions contributes to a deeper understanding of the equation’s dynamics, particularly in real-world applications such as quantum optics and condensed matter physics. Additionally, this study incorporates graphical depictions of specific solutions to demonstrate the effect of white noise on solitons visually.

Efficient simulation of complex Ginzburg–Landau equations using high-order exponential-type methods

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies.

Stability of bright solitons in optical system supported by cubic and quintic nonlinearities with PT -symmetric quartic harmonic complex potential

This work analyzed the existence and stability of optical waves in linear and nonlinear media with self-focusing cubic-quintic nonlinearity by way of interactions of nonlinearity, spatial diffraction, and parity-time ( PT ) symmetric complex potential defined by quartic harmonic real potential and Gaussian imaginary distribution. We demonstrated that the order of nonlinear function, the width and depth of the real potential, and the coefficient of diffraction modify the span of the complex potential parameter onto which PT -symmetry breakdown occurs. The region of stable PT -symmetry in the self-focusing medium improves as the order of the nonlinear function increases, as demonstrated by comparing the soliton solutions with cubic and quintic nonlinearities. We addressed the nonlinear evolution of the stable and unstable PT solitons under propagation. The linear stability of bright solitons is extensively studied and established the conditions for stable soliton for both PT− symmetric and broken PT− symmetric regions.

Analytical and numerical solutions to the generalized Schrödinger equation with fourth-order dispersion and nonlinearity

This study aims to solve the generalized Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity (4-Order-GNLSE) using advanced analytical and numerical methods. The 4-Order-GNLSE is critical in understanding complex wave propagation phenomena in various physical contexts, including optical fibers and fluid dynamics, where higher-order dispersion and nonlinear effects are significant. We employ the Khater II (Khat II) and modified Rational (MRat) methods to construct analytical solutions. To validate these solutions, we utilize the trigonometric-quantic-B-spline (TQBS) method as a numerical scheme, demonstrating a close match between analytical and numerical results. Our findings indicate that the proposed methods effectively capture the intricate dynamics governed by the 4-Order-GNLSE. The results highlight the relevance of these solutions in practical applications, offering new insights into the wave propagation behaviors influenced by higher-order dispersion and nonlinearities. This research provides a significant contribution to the field of nonlinear wave equations, presenting novel solutions and validating methodologies that enhance our understanding and potential applications of these complex systems.

Motion dynamics of two-dimensional fundamental and vortex solitons in the fractional medium with the cubic-quintic nonlinearity

Motion dynamics of two-dimensional fundamental and vortex solitons in the fractional medium with the cubic-quintic nonlinearity

Exact solutions to the fractional complex Ginzburg-Landau equation with cubic-quintic and Kerr law nonlinearities

The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved (G′/G2) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.

Multiple solitons structures in optical fibers via PNLSE with a novel truncated M-derivative: modulated wave gain

This study introduces a novel truncated Mittage–Leffler (M)- proportional derivative (TMPD) and examines its impact on the perturbed nonlinear Schrödinger equation (PNLSE) that includes fourth-order dispersion and cubic-quintic nonlinearity. The TMPD-PNLSE is used to model light signals in nanofibers. In addition to dispersion and Kerr nonlinearity, which are characteristics of the NLSE, the PNLSE also exhibits self-steepening and self-phase modulation effects. The unified method is implemented to derive exact solutions for the model equation. These solutions provide a variety of phenomena; including breathers, geometric chaos, and complex solitons. The solutions also exhibit numerous structures, such as geometric chaos, where undulated M-shaped and M-shaped solitons are embedded. The modulation instability is analyzed, finding that it is triggered when the coefficient of the fourth-order dispersion surpasses a critical value.

The periodic soliton solutions for a nonlocal nonlinear Schrödinger equation with higher-order dispersion

A nonlocal nonlinear Schr¨odinger (NNLS) equation with fourth-order dispersion and cubic-quintic nonlinearities has been studied analytically and numeri- cally. Under the constraint conditions, auxiliary functions are introduced, and explicit one- and two-soliton solutions are obtained by the Hirota bilinear method. Accord- ing to the solutions, the propagation dynamics of soliton pulses are investigated. The influences of different parameters on the dynamics of one- and two-soliton solutions have been analyzed. The results show that the two-soliton solution exhibits diverse dy- namic characteristics under the suitable parameter selections. In addition, the stability of one- and two-soliton solutions against the constraint conditions deviations and under the initial perturbations are also studied numerically.

Stochastic higher-order Lakshmanan–Porsezian–Daniel model with cubic-quintic law nonlinearities in the presence of spatio-temporal and chromatic dispersion terms

This work focuses on deriving optical soliton solutions for the stochastic variant of the higher-order Lakshmanan–Porsezian–Daniel (LPD) model, which incorporates cubic-quintic law nonlinearities alongside chromatic and spatio-temporal dispersion terms. The primary approach chosen for this purpose is the new Kudryashov method (nKM). Initially, bright and dark soliton solutions are derived. The stability of these dark and bright solitons is then verified using the Vakhitov–Kolokolov criterion. Subsequently, the impact of the stochastic term on the bright and dark solitons is examined. As an additional aspect, an investigation has been conducted on the effects of the model parameters, namely, group velocity dispersion and spatio-temporal dispersion terms.

Exact solutions to a family of nonlinear Schrödinger equations

In this article, using the method of differential constraints and Lie group analysis, new exact kink-like solutions are obtained for certain families of nonlinear Schrödinger equations with cubic-quintic nonlinearity. The foregoing solutions are presented in terms of the Lambert W function.

Transformation of rotating dipole and vortex solitons in an anharmonic potential

We investigate the basic properties of dipole and vortex solitons, both quiescent and rotating, supported by the axially symmetric anharmonic potential in a medium with cubic-quintic nonlinearity. The static solitons exhibit nonmonotonous behavior in terms of propagation constant as a function, and feature bistability regions. In the rotating frame, the dipole solitons transform into vortex solitons and the topological charge increases gradually with the growth of rotation frequency. Linear stability analysis and direct simulations reveal that rotating dipole solitons with higher charges are more stable than those with lower charges. Meanwhile, dipole solitons can rotate persistently during propagation and preserve their shape over multiple rotation periods. Our findings provide an alternative way for the transformation to achieve dipole and vortex solitons with higher topological charges.

Modulation instability, bifurcation analysis and solitonic waves in nonlinear optical media with odd-order dispersion

In this paper, modulation instability, bifurcation analysis, and soliton solutions are investigated in nonlinear media with odd-order dispersion terms. A generalized nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity is considered. A linear analysis method is used to derive an expression of the modulation instability spectrum, and the effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. The results show that for negative values of the fourth order in a normal dispersion regime, the modulation instability vanishes. Another important aspect is revealed when the fourth-order dispersion gets positive and has high values; additional bands emerge to enlarge the bandwidths of the modulation instability. To further confirm the effects of the odd-order dispersion in the nonlinear structure, the bifurcation, phase portraits, and chaotic behaviors have been investigated to show how instability arises in the nonlinear structure. Using numerical simulation, in particular the Runge-Kutta algorithm, the sensitivity of nonlinear systems is pointed out to confirm an unstable behavior. This investigation confirms once again the fact that the modulation spectrum is sensitive to higher-order dispersion in normal and anomalous dispersion regimes. A traveling wave hypothesis is employed to lead to the direct integration of the nonlinear system, and some specific soliton solutions are extracted. For particular constraint conditions on the discriminant, bright and dark solitons as well as Jacobi elliptic function solutions emerge. The obtained results could be used to improve the transmission signal via the optical fiber and secure the data in communication systems.

Multi-stable multipole solitons in competing nonlinearity media

We investigate the properties of two-dimensional multipole solitons supported by the harmonic-Gaussian potential in a medium with competing cubic-quintic nonlinearity. The combined potential provides a shallow radial minimum for trapping solitons. We find that the multipole solitons with different power can propagate stably in some of the same parameter domains. Meanwhile, the solitons have multiple stable domains at the higher power, i.e., multi-stable multipole solitons. Linear stability analysis and numerical simulations show that the stability domain of solitons shrinks gradually with the increasing number of bright spots. Our findings suggest an alternative way for the realization of stable multipole solitons.

Exact soliton solutions of a (2+1)-dimensional time-modulated nonlinear Schrödinger equation with cubic-quintic nonlinearity

In the present paper, based on the wave ansatz method, exact bright and dark solitons of a (2+1)-dimensional time variable coefficients nonlinear Schrödinger equation with cubic-quintic nonlinearity term are presented. This model can be used to describe the propagation of optical pulse in inhomogeneous media. These obtained solitons exhibit nonautonomous properties due to the amplitude and velocity of them are time-dependent coefficients. At this time, Tangent and Secant function solutions are also reported.

Discrete solitons in competitive zigzag waveguide arrays with cubic-quintic nonlinearity

In this paper, we study one-dimensional discrete solitons in zigzag waveguide arrays with competitive cubic-quintic nonlinearity and competitive linear mixing between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. The competitive nonlinearity features a cubic self-focusing associated with a quintic self-defocusing nonlinearities. The competitive linear mixing between the NN and NNN couplings is induced by making the two coefficients opposite of each other, which is assumed to be induced by the embedding synthetic gauge phase within the coupling constants. The combination of these two types of competition, linear mixing and nonlinearity can create four types of soliton: multipeak bell-shaped solitons, multipeak flat-top solitons, staggered bell-shaped solitons, and staggered flat-top solitons. The stability and dynamics of these types of solitons are verified systematically through the paper. The total power and the types of competition between the linear mixing play important roles in tuning these solitons.