Fractional Temporal Evolution Research Articles

Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.

Optical soliton perturbation with fractional temporal evolution having cubic and power law nonlinearities

With an increasing demand for larger bandwidth and solving the problem of internet bottleneck, the formation of solitons with fractional temporal evolution has prompted substantial attention in the field of optics. In this paper, we have used three methods namely, the new mapping method, the unified Riccati equation method, and the new modified sub-ODE method to construct the new soliton solutions to the perturbed nonlinear Schrödinger equation (NLSE) with fractional temporal evolution. Two types of nonlinearities are considered namely, the Kerr law and the power law for all the three utilized methods. Existence criteria for all soliton solutions are presented. We provide two- and three-dimensional graphical illustrations of some acquired solutions. This work also proves the reliability, power, and effectiveness of these new analytical techniques for solving other nonlinear fractional differential equations.

Doubly periodic wave structure of the modified Schrödinger equation with fractional temporal evolution

Abundant Jacobi elliptic type solutions with distinct physical structures of complex nonlinear conformable time-fractional modified Schrödinger equation are obtained by using the generalized Jacobi elliptic function (GJEF) method. The Jacobi function expansions may lead to new doubly periodic wave solutions, soliton solutions, and triangular periodic solutions. Nowadays the conformable operator is being used for a better description of the dynamical systems. Motivated by the potential applications of the governed equation in nonlinear optics, biological sciences, and fluid dynamics, these solutions may be significant in the study of wave propagation in the desired field. Symbolic computations are made with the aid of Maple.

Extraction of New Wave Sturctures Related to Some Importantmodels in Shallow Water Waves with Fractional Temporalevolution

Extraction of New Wave Sturctures Related to Some Importantmodels in Shallow Water Waves with Fractional Temporalevolution

On analytic series solutions and conserved fluxes of the time fractional (2+1)-dimensional Burger's system via invariant approach

The article investigates some new properties such as invariant transformations, conserved vectors, and closed-form series solutions of (2+1) dimensional nonlinear coupled Burger system, a model of the evolution of the scaled volume concentration or sedimentation of the two kinds of particles in fluid suspensions of colloids, under the effect of gravity, with fractional temporal evolution by the Lie symmetry method. It is worth noting that the series solutions of (2+1) or higher-dimensional system is not much explored. Moreover, nonlocal conservation laws, one of the important aspects of symmetries, are constructed by the generalized Noether theorem.

Effects of fractional temporal evolution on chirped soliton solutions of the Chen-Lee-Liu equation

In this paper, we show the effect of fractional temporal evolution on chirped soliton solutions of the Chen-Lee-Liu equation (CLLE). We adopt the New Modified SARDAR Sub-equation Method to derive bright and dark solitons, periodic and singular function solutions, and plot some of them in (3D) dimension. We prove that smaller is the order of the temporal derivative, greater is the effect on the propagating waves. This characterization gives better and full information about these propagating waves, and therefore, it leads to a marked improvement in the multiple related relevant applications.

Optical Soliton perturbation with fractional temporal evolution by extended modified auxiliary equation mapping

In this article we discussed the analytical analysis of perturbed nonlinear fractional Schr$\ddot{o}$dinger equation with the help of our new introduced technique extended modified auxiliary equation mapping method(EMAEMM). By using this newly proposed technique we have found some new and more general variety of exact traveling wave solutions which are collecting some kind of semi half bright, semi half dark, doubly periodic, dark, combined, periodic, bright, half dark and half bright via three parametric values which is the primary key point of difference of our technique. These results are highly applicable to develop new theories of plasma physics, quantum mechanics, biomedical problems, soliton dynamics, nuclear physics, optical physics, fluid dynamics, electromagnetism, industrial studies, mathematical physics, and in many other natural and physical sciences. For detailed physical dynamical representation of our results we have shown them with graphs in different dimensions via Mathematica 10.4 to get more understanding to observe the behavior of different new dynamical shapes of solutions.

Optical solitons with fractional temporal evolution in fiber bragg gratings with generalized anti-cubic nonlinearity by the fractional Riccati method

This paper presents dark optical and other solutions for generalized anti-cubic (AC) nonlinearity relevant to fiber Bragg gratings (BGs), Including fractional temporal evolution. The concept of modified Riemann-Liouvill derivative of Jumarie of order α, coupled with the method of fractional Riccati, is used. For its nature, the soliton solution appears with restriction conditions.

Abundant fractional solitons to the coupled nonlinear Schrödinger equations arising in shallow water waves

In this work, the dynamics of wave phenomena modeled by (2[Formula: see text]+[Formula: see text]1)-dimensional coupled nonlinear Schrodinger’s equations with fractional temporal evolution is studied. The solutions of the equations are two monochromatic waves with nonlinear modulations that have almost identical group velocities. The unified approach along with the properties of the local M-derivative are used to obtain dark and rational soliton solutions. The restrictions on parameters ensure that these soliton solutions are persevering. Lastly, the influence of the fractional parameter upon the obtained results are evaluated and depicted through graphs.

Exact solutions and bifurcation for the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution

ABSTRACT The resonant nonlinear Schrödinger equation (RNLSE) with competing weakly nonlocal nonlinearity and fractional temporal evolution, which describes the propagation of optical solitons along the nonlinear optical fibers, is investigated in this paper. Dynamic behavior of such equation with the parabolic-type nonlinearity is discussed. The relationship between the orbits of the dynamic system and traveling wave solutions is demonstrated. Particularly, a family of hyperbolic curves near the saddle points implies the existence of one family of the braking wave solutions correspondingly. Furthermore, the -expansion method and Jacobi elliptic function rational expansion method are conducted to get the exact traveling wave solutions, such as periodic wave solutions, shock wave solutions, and breaking wave solutions.

Oblique plane waves with bifurcation behaviors and chaotic motion for resonant nonlinear Schrodinger equations having fractional temporal evolution

This study deal with the oblique plane wave solutions with dynamical behaviours for (2 + 1)-dimensional resonant nonlinear Schrodinger equations having Bhom’s quantum potential with distinct law of nonlinearities (Kerr and parabolic law) and fractional temporal evolution. The considered equations are converted to solvable form by assuming conformable Khalil’s fractional derivatives. The bifurcation behaviors and chaotic motion for the existence of traveling waves are investigated by forming the planar dynamical system from the considered equations. The novel auxiliary ordinary differential equation method is therefore used to divulge several forms of plane wave solutions of these equations. It is investigated that the widths of resonant wave dynamics are significantly modified with the influence of obliqueness. The evaluated results may be very useful for better examining the resonant optical solitons in nonlinear dynamics because of obliqueness are existed in various nonlinear systems, specifically in optical bullets, Madelung fluids, etc.

Oblique resonance wave phenomena for nonlinear coupled evolution equations with fractional temporal evolution

This work investigates the obliquely propagating resonance wave phenomena described by coupled model equations with fractional temporal evolution arising in many branches of physics. The (2 + 1) -dimensional coupled nonlinear Schrodinger equation (NLSE), long wave-short wave resonance interaction (LSRI) equation and Maccari system (MS) having fractional time evolution are considered to study such physical phenomena. The modified Kudryashov method (mKM) with the properties of Khalil's conformable fractional derivatives is implemented to divulge scattered oblique wave solutions of considered equations. The effects of obliqueness and fractional parameter on obtained results are demonstrated graphically along with the physical descriptions. It is found that the resonance wave phenomena are changed with the increase of obliqueness as well as fractional parameter.

Bifurcation of traveling waves and exact solutions of Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution

The article is devoted in the study of the Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution in which modified Riemann–Liouville derivative is considered. We discuss the dynamical behavior of the system using bifurcation theory in different parametric regions. We also depict the phase portraits of the traveling wave solutions and obtain explicit traveling waves including solitary wave and breaking wave solution. We apply the $$(G'/G)$$ -expansion method and the F-expansion method along with fractional complex transformation, and obtain a large variety of exact traveling wave solutions which includes solitary wave, kink-type wave, and periodic wave solutions of the equation. Finally, we demonstrate remarkable features of the traveling wave solutions via interesting figures and phase portraits.

Resonant optical solitons with fractional temporal evolution by modified extended direct algebraic method

This paper employs modified extended direct algebraic method to recover bright, dark and singular solitons for resonant nonlinear Schrödinger's equation that is studied with dual-power law media. Singular periodic solutions also emerge as a byproduct of this integration scheme.

Suppressing internet bottleneck with fractional temporal evolution of cubic–quartic optical solitons

This paper obtains bright, dark and bright–dark combo cubic–quartic optical solitons when group velocity dispersion is negligible or totally absent. The model is studied with fractional temporal evolution by F-expansion scheme with Kerr and parabolic laws of nonlinearity. The results can be used to achieve slow-light dynamics thus enabling the control of Internet traffic flow.

Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−Φ(ξ))-expansion

This work studies fractional temporal evolution of oblique resonant optical solitons in (3+1)-dimensions with Kerr- and parabolic-law nonlinearities. The generalized exp(−Φ(ξ))-expansion method along with the Khalil's conformable fractional derivatives is implemented to locate several forms of oblique resonant solitons. It is observed that obliqueness significantly modified resonant wave dynamics. The obtained results are very useful for understanding the dynamics of obliquely propagating resonant optical solitons and optical bullets.

Optical solitons with quadratic–cubic nonlinearity and fractional temporal evolution

The paper studies the single and combined optical solitons in the physical model of telecommunication system that describes the dynamics of optical solitons with quadratic–cubic nonlinearity in the mitigate internet bottleneck. This fractional temporal evolution is considerd with the coefficients of self-steeping, group velocity, inter-model and nonlinear dispersions. The extended Fan sub-equation method is used to investigate the families of single optical solitons (bright, dark and singular) for the different values of parameters.

Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution

This paper secures dark and singular resonant optical solitons that is studied with dual-power law nonlinearity and fractional temporal evolution. Khalil's conformable fractional derivative is put into perspective to retrieve these soliton solutions. The parameter restrictions for the existence of such solitons are also indicated.

Optical soliton perturbation with fractional temporal evolution by generalized Kudryashov's method

This paper retrieves optical soliton solutions with fractional temporal evolution by the aid of generalized Kudryashov's method. There are four types of nonlinear fibers that are studied here. Bright, dark and singular soliton solutions are retrieved. The existence criteria of these solitons are also presented.

Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic–cubic nonlinearity

This paper studies fractional temporal evolution of optical solitons with quadratic–cubic nonlinearity that comes with a few perturbation terms. Khalil's conformable fractional derivative as well as Liu's extended trial function scheme are applied to retrieve these soliton solutions. The results are applicable to mitigate Internet bottleneck that is a growing problem in telecommunications industry.