On soliton solutions for Boussinesq–Burgers equations

Rady, A.S. Abdel; Khalfallah, Mohammed

doi:10.1016/j.cnsns.2009.05.039

Cited by 35 publications

(10 citation statements)

References 17 publications

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“…Essentially, the constraint equation is related to linear heat equation which result in new exact solutions of the Boussinesq-Burgers equation. These solutions are different from those solutins given in literatures [ 26 – 32 ]. It shows that our method is more flexible in finding more general form exact solutions and the method can be used for many other NLEEs in mathematical physics.…”

Section: Discussioncontrasting

confidence: 81%

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A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

Bilige

Wang

2015

PLoS ONE

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In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

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Section: Discussioncontrasting

confidence: 81%

“…In this section, we are aimed to first give the most general form exact solutions, then we will determine the exact traveling wave and non-traveling wave solutions for the Boussinesq-Burgers equation [ 26 – 32 ] …”

Section: Application Of the Methodsmentioning

confidence: 99%

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

Bilige

Wang

2015

PLoS ONE

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“…But now the interest is increasing to study localized soliton solutions in higher dimensions. Numerous techniques, including the homogenous balance method [13][14][15][16][17][18], the hyperbolic function expansion method [21,22], the sine-cosine method [23], the nonlinear transformation method [24][25][26], and the trial function method [27,28], have been proposed in order to obtain the periodic wave and soliton solutions of the nonlinear evolution equations. The generalised periodic solutions cannot be derived using these approaches, which can only produce shock or solitary wave solutions, or periodic wave solutions in terms of basic functions.…”

Section: Introductionmentioning

confidence: 99%

The investigation of dynamical behaviour variation of (2+1)-dimensional Extended Calogero-Bogoyavlenskii-Schiff Equation with different fractional operators

Mahmood

Abbas

Huntul

et al. 2023

Preprint

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The majority of scientific fields employ differential equations involving fractional-order derivatives to understanda wide range of physical events. Fractional derivatives and integrals have the ability to cop with complicatedproblems. Various types of fractional-order derivative have been dicovered, like -fractional and M-truncatedderivative. In this article the Extended Calogero-Bogoyavlenskii-Schiff Equation (CBSE) is used in its fractionalform to extract the soliton solutions, as well as to observe the behaviour of -fractional and M-truncated derivative.CBSE equation has been used to analyse various events in many fields of research. The analytical solutionsare drawn by using the Jacobi elliptic function (JEF) and the expanded tanh-coth techniques. These analyticalsolutions exhibit solitary and periodic wave behaviours. The perposed techniques are one of the effective methodsto obtain soliton solutions of partial differential equations (PDEs). The results are depicted graphically usingdifferent parametric values and the fractional parameters. The wave behaviour is observed, and shown by 3D and 2D plots.

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“…Moreover, the bilinear form and soliton solutions were provided via the Hirota method and the binary Bell polynomials by Wang et al [35]. Jacobi elliptic function method was used to construct periodic and multiple soliton solutions by Rady and Khalfallah [36].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions and bifurcations of the time-fractional coupled Boussinesq-Burgers equation

Liu,

Xu,

Wang

2023

Phys. Scr.

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In this paper, the time-fractional coupled Boussinesq-Burgers equations are studied to construct exact solutions by utilizing the modified extended tanh-function method, the improved (G'/G) method, the two variables (G'/G,1/G)-expansion method and bifurcation analysis. Firstly, based on the modified Riemann-Liouville derivative and traveling wave transformation, the equations are rewritten as an ordinary differential equation. Some exact solutions, including singular, periodic, kink, anti-kink, soliton and periodic-singular solutions, are derived. Then, we graphically describe the 3D, density plots and solution profiles of some solutions by setting suitable parameters. Finally, the effectiveness of the methods applied is verified via bifurcation theory.

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On soliton solutions for Boussinesq–Burgers equations

Cited by 35 publications

References 17 publications

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

The investigation of dynamical behaviour variation of (2+1)-dimensional Extended Calogero-Bogoyavlenskii-Schiff Equation with different fractional operators

Exact solutions and bifurcations of the time-fractional coupled Boussinesq-Burgers equation

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