Double-wave solutions and Lie symmetry analysis to the (2 + 1)-dimensional coupled Burgers equations

Osman, M. S.; Băleanu, Dumitru; Adem, Abdullahi Rashid; Hosseini, K.; Mirzazadeh, Mohammad; Eslami, Mostafa

doi:10.1016/j.cjph.2019.11.005

Cited by 115 publications

(20 citation statements)

References 22 publications

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“…It can be said that the Lie symmetry method is the most important approach for constructing analytical solutions of nonlinear PDEs. It is based to study the invariance of differential equations (DEs) under a one-parameter group of transformations which transforms a solution to another new solution and is also used to reduce the order such as the number of variables of DEs; moreover, the conservation laws (CLs) can be constructed by using the symmetries of the DEs (see [19][20][21][22][23][24]). A short time ago, the Lie symmetry analysis is also used for FDEs; in [25], Gazizov et al showed us how the prolongation formulae for fractional derivatives is formulated; by this work, the Lie group method becomes Valid for FDEs; after that, many researches are devoted for studying the FDEs by using Lie symmetry analysis method, for more details see ( [26][27][28][29]).…”

Section: Introductionmentioning

confidence: 99%

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

Bahi

Hilal

2021

Journal of Function Spaces

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In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

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Section: Introductionmentioning

confidence: 99%

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

Bahi

Hilal

2021

Journal of Function Spaces

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“…On one hand, since integrability of the fractional models is much poorer than that of the classical ones, the compound function solutions of (1.2), for typical traveling wave solutions, were hardly obtained by adopting some direct methods. On the other hand, there have been abundant studies on Lie symmetries, conservation laws, and exact explicit solutions for many integer and fractional real PDEs [22][23][24][25][26][27][28][32][33][34][35][36][37][38][39][40][41][42][43]. However, few symmetries of the time-fractional complex system have been discussed until now, even non-homogenous ones.…”

Section: Introductionmentioning

confidence: 99%

Non-classical symmetry and analytic self-similar solutions for a non-homogenous time-fractional vector NLS system

Ren

Zhang

2021

Adv Differ Equ

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The complex PDEs are a very important and interesting task in nonlinear quantum science. Although there have been extensive studies on the classical complex models, solving the fractional complex models still has a lot of shortcomings, especially for the non-homogenous ones. Therefore, the present study focuses on solving the two-component non-homogenous time-fractional NLS system, our method is to solve a prolonged fractional system derived from the governed model. We first establish non-classical symmetries of this new enlarged system by using the fractional Lie group method. Then, with the help of fractional Erdélyi–Kober operator, we reduce this new system into fractional ODEs, the self-similar solutions are obtained via the power series expansion. The convergence of these solutions are proven as all the variable coefficients are analytic. Finally, we generalize our methods to handle the multi-component case. We conclude that this way may also bring some convenience for solving other complex systems.

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“…It is known that these exact solutions of nonlinear evolution equations (NLEEs), especially the soliton solutions [1][2][3], can be given by using a variety of different methods [4,5], such as Jacobi elliptic function expansion method [6], inverse scattering transformation (IST) [7,8], Darboux transformation (DT) [9], extended generalized DT [10], Lax pair (LP) [11], Lie symmetry analysis [12], Hirota bilinear method [13], and others [14,15]. The Hirota bilinear method is an efficient tool to construct exact solutions of NLEEs, and there exists plenty of completely integrable equations which are studied in this way.…”

Section: Introductionmentioning

confidence: 99%

Application of the Multiple Exp-Function, Cross-Kink, Periodic-Kink, Solitary Wave Methods, and Stability Analysis for the CDG Equation

Jebreen

Chalco-Cano

2021

Advances in Mathematical Physics

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In this article, the exact wave structures are discussed to the Caudrey-Dodd-Gibbon equation with the assistance of Maple based on the Hirota bilinear form. It is investigated that the equation exhibits the trigonometric, hyperbolic, and exponential function solutions. We first construct a combination of the general exponential function, periodic function, and hyperbolic function in order to derive the general periodic-kink solution for this equation. Then, the more periodic wave solutions are presented with more arbitrary autocephalous parameters, in which the periodic-kink solution localized in all directions in space. Furthermore, the modulation instability is employed to discuss the stability of the available solutions, and the special theorem is also introduced. Moreover, the constraint conditions are also reported which validate the existence of solutions. Furthermore, 2-dimensional graphs are presented for the physical movement of the earned solutions under the appropriate selection of the parameters for stability analysis. The concluded results are helpful for the understanding of the investigation of nonlinear waves and are also vital for numerical and experimental verification in engineering sciences and nonlinear physics.

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Double-wave solutions and Lie symmetry analysis to the (2 + 1)-dimensional coupled Burgers equations

Cited by 115 publications

References 22 publications

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

Non-classical symmetry and analytic self-similar solutions for a non-homogenous time-fractional vector NLS system

Application of the Multiple Exp-Function, Cross-Kink, Periodic-Kink, Solitary Wave Methods, and Stability Analysis for the CDG Equation

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