Analytical study of exact traveling wave solutions for time-fractional nonlinear Schrödinger equations

Ilie, Mousa; Biazar, Jafar; Ayati, Zainab

doi:10.1007/s11082-018-1682-y

Cited by 18 publications

(4 citation statements)

References 45 publications

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“…The second problem is the nonlinear fractional Phi-4 equation for which the (G /G, 1/G)-expansion method will be used. Some recent work utilizing the conformable fractional derivative in real world problems such the Burgers-KdV equation, the KdV-Zakharov-Kuznetsev equation, the unstable Schrödinger equation, and the resonant nonlinear Schrödinger equation have been studied in [28][29][30]. In addition, exact solutions to equations associated with the above two problems have been found using many methods.…”

Section: Introductionmentioning

confidence: 99%

Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods

et al. 2019

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Ultrashort pulse propagation in optical transmission lines and phenomena in particle physics can be investigated via the cubic-quintic Ginzburg-Landau equation and the Phi-4 equation, respectively. The main objective of this paper is to construct exact traveling wave solutions of the (2 + 1)-dimensional cubic-quintic Ginzburg-Landau equation and the Phi-4 equation of space-time fractional orders in the sense of the conformable fractional derivative. The method employed to solve the Ginzburg-Landau equation and the Phi-4 equation are the modified Kudryashov method and the (G /G, 1/G)-expansion method, respectively. Several types of exact analytical solutions are obtained including reciprocal of exponential function solutions, hyperbolic function solutions, trigonometric function solutions and rational function solutions. Graphical representations and physical explanations of some of the obtained solutions are demonstrated using a range of fractional orders. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. These methods are simple and efficient for solving the proposed equations.

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Section: Introductionmentioning

confidence: 99%

Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods

et al. 2019

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“…Fractional nonlinear evolution equations (FNLEEs), are mathematical models that describe many complicated phenomena in nature and science particularly, fiber-optic communication, transport processes, plasma physics, bio-engineering, signal processing, fluid dynamics, control theory, fractal theory, nuclear reactor kinetics, electrical circuits, and viscoelastic materials, etc [1,2]. Some noteworthy FNLLEs in the aforementioned domains are the family of Schrodinger equations [3][4][5], the Heisenberg ferromagnetic spin chain [6], Kundu-Eckhaus equation [7], the reaction-diffusion equation [8], the NPCLL equation [9], The Boussinesq equation [10], Kadomtsev-Petviashvili equation [11], The Klein-Gordon equation [12], The Zakharov-Kuznetsov (ZK) equation [13], and so on. The analytical solution of these equations piqued the interest of both theoretical and experimental researchers in the last few decades.…”

Section: Introductionmentioning

confidence: 99%

“…Based on the homogeneous balance principle of soliton theory, numerous techniques have been discovered. For instance, the Sine-Gordon expansion scheme [5], the auxiliary equation method [6], the tanh-coth method [14], the modified -expansion and modified Kudryashov methods [15], the modified -expansion function strategy [9], the Kudryashov scheme [16], the extended Kudryashov method [17], and so on. The analytical solutions derived using the methods described in the above are a sort of traveling wave solution known as a soliton solution.…”

Section: Introductionmentioning

confidence: 99%

Diverse optical soliton solutions of two space-time fractional nonlinear evolution equations by the extended kudryashov method

Devnath,

Akbar,

Gómez-Aguilar

2023

Phys. Scr.

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This study investigates the comprehensive optical soliton solutions to the (2+1)-dimensional nonlinear time-fractional Zoomeron equation and the space-time fractional nonlinear Chen-Lee-Liu equation using the extended Kudryashov technique. The newly defined beta derivative is used to conduct the fractional terms and investigate wide-spectral soliton solutions to the considered models. The general solutions yield a variety of typical soliton shapes, including V-shaped soliton, anti-bell-shaped soliton, kink soliton, periodic soliton, singular periodic soliton, etc. The three-dimensional, contour, and two-dimensional graphs of the derived solitons have been plotted to illustrate the structure, propagation, and influence of the fractional parameter. The precision of the acquired solutions is confirmed by reintroducing them into the original equation using Mathematica. The findings of this study indicate that the employed method has the capability of yielding compatible, creative, and useful solutions for diverse nonlinear evolution equations with fractional derivatives. This approach could introduce novel ways for unraveling other nonlinear equations and have implications in diverse sectors of nonlinear science and engineering.

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“…The authors in [17] introduced a useful conformable derivative; in addition, a non-conformable derivative is introduced in [18]. These derivatives are interesting from a theoretical viewpoint and useful in many applications [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

New Hermite–Hadamard Type Inequalities Involving Non-Conformable Integral Operators

2019

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Analytical study of exact traveling wave solutions for time-fractional nonlinear Schrödinger equations

Cited by 18 publications

References 45 publications

Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods

Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods

Diverse optical soliton solutions of two space-time fractional nonlinear evolution equations by the extended kudryashov method

New Hermite–Hadamard Type Inequalities Involving Non-Conformable Integral Operators

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