Multi-soliton solution, rational solution of the Boussinesq–Burgers equations

Rady, A.S. Abdel; Osman, El-Sayed; Khalfallah, Mohammed

doi:10.1016/j.cnsns.2009.05.053

Cited by 22 publications

(9 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These functions encompass special cases like z h sec and z tanh , which provide explanations for the propagation of solitary and shock waves. The values of the constants a i (where i = 0, 1, 2,K, n) in equation (7) depend significantly on the nature of the involved partial differential equations. Different solution forms can be categorized using d 0 , d 1 , d 2 , d 3 , and d 4 , and classified into cases 1-3.…”

Section: Overview Of the Jacobi Elliptic Function And Conformable Fra...mentioning

confidence: 99%

“…Moreover, through the substitution of the obtained solutions for the set of algebraic equations into equation (7), along with the incorporation of solutions from equation (8), we can establish…”

Section: Overview Of the Jacobi Elliptic Function And Conformable Fra...mentioning

confidence: 99%

“…Several techniques have been proposed to obtain periodic wave and soliton solutions for nonlinear evolution equations. These methods include the homogenous balance method [1][2][3][4][5][6][7][8][9], the hyperbolic function expansion method [10,11], the sine-cosine method [12], the nonlinear transformation method [13][14][15], and the trial function method [16,17]. However, these approaches are limited to yielding shock or solitary wave solutions or periodic solutions expressed in terms of elementary functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

Elsadany,

Alshammari,

Elboree

2024

Phys. Scr.

View full text Add to dashboard Cite

In this research, we investigate the e ects of fractional order on the (3 + 1)-dimensional general-ized space-time fractional modi ed KdV-Zakharov-Kuznetsov (mKdV-ZK) equation. We approachthe problem by utilizing the conformable fractional derivative. By reducing the mKdV-ZK equationto an integer order nonlinear ordinary di erential equation, we apply the Jacobi elliptic functionmethod to nd exact solutions. These solutions are speci cally tailored for the fractional order ofthe (3 + 1)-dimensional generalized mKdV-ZK equation, encompassing solitary waves, shock waves,and periodic waves. We also compare these exact solutions with fractional solutions to gain furtherinsights. Notably, our approach demonstrates the feasibility of solving nonlinear time-fractionaldi erential equations with conformable derivatives. Several diagrams have been included to visuallydepict the behavior of the solutions under fractional order when certain special parameter valuesare employed.

show abstract

Section: Overview Of the Jacobi Elliptic Function And Conformable Fra...mentioning

confidence: 99%

Section: Overview Of the Jacobi Elliptic Function And Conformable Fra...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

Elsadany,

Alshammari,

Elboree

2024

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…For system(1), finite-band solutions have been investigated with the help of the Lax representations of some stationary evolution equations [14], rational solutions have been obtained via the bilinear method and Kadomtsev-Petviashvili hierarchy reduction [15], non-auto-Bäcklund transformations, nonlocal symmetries, conservation laws and interaction solutions have been constructed [16], Darboux transformations and multi-soliton solutions have been presented [17,18], bilinear form and multi-soliton solutions have been studied [19]. By the way, other issues related to system (1) have been seen [20][21][22][23][24][25][26][27][28].…”

mentioning

confidence: 99%

Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Gao¹,

Guo²,

Shan³

2020

Commun. Theor. Phys.

View full text Add to dashboard Cite

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-Bäcklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and Bäcklund transformations are different from those in the existing literature. All of our results are dependent on the water-wave dispersive power.

show abstract

Multi-soliton solution, rational solution of the Boussinesq–Burgers equations

Cited by 22 publications

References 14 publications

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

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

Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Contact Info

Product

Resources

About