New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water

Yan, Zhenya; Zhang, Hongqing

doi:10.1016/s0375-9601(01)00376-0

Cited by 322 publications

(120 citation statements)

References 11 publications

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“…When γ 1 (t) = γ 2 (t) = γ 4 (t) = 1, γ 3 (t) = γ 5 (t) = β and γ 6 (t) = α are all constants, equations (1.1) and (1.2) transform into the WBK equations in shallow water [52]:…”

Section: Introductionmentioning

confidence: 99%

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

Zhang¹,

Wang²

2017

J. Nonlinear Sci. Appl.

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In this paper, Whitham-Broer-Kaup (WBK) equations with time-dependent coefficients are exactly solved through Hirota's bilinear method. To be specific, the WBK equations are first reduced into a system of variable-coefficient Ablowitz-KaupNewell-Segur (AKNS) equations. With the help of the AKNS equations, bilinear forms of the WBK equations are then given. Based on a special case of the bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions and the uniform formulae of n-soliton solutions are finally obtained. It is graphically shown that the dynamical evolutions of the obtained one-, two-and three-soliton solutions possess time-varying amplitudes in the process of propagations.

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“…When γ 1 (t) = γ 2 (t) = γ 4 (t) = 1, γ 3 (t) = γ 5 (t) = β and γ 6 (t) = α are all constants, equations (1.1) and (1.2) transform into the WBK equations in shallow water [52]:…”

Section: Introductionmentioning

confidence: 99%

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

Zhang¹,

Wang²

2017

J. Nonlinear Sci. Appl.

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“…In recent years, several methods have been established for investigating nonlinear PDEs to obtain exact solutions. For example, the Hirota's bilinear transformation method [1], the B cklund transformation method [2], the Jacobi elliptic function expansion method [3], the generalized Riccati equation method [4,5], the homogeneous balance method [6], the F-expansion method [7], the Exp-function method [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Some new solutions of the (1+1)-dimensional PDE via the improved (G′/G)-expansion method

Naher¹,

Abdullah²

2014

AIP Conference Proceedings

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Abstract. In this article, an improved '/ G G -expansion method is implemented for the simplified Modified CamassaHolm (MCH) equation involving parameters, with an aim to construct many new traveling wave solutions. In this method, second order linear ordinary differential equation with constant coefficients has been implemented as an auxiliary equation. The generated solutions including solitons and periodic solutions are demonstrated by the hyperbolic function, the trigonometric function and the rational forms. If the parameters take particular values, the solutions become in special functional form. Moreover, it is worth mentioning that, some of our solutions are in good agreement with already published results in the open literature by setting appropriate values of constants, which proves our other solutions. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple 13.

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“…So, finding exact solutions is important to understand the mechanism of the complicated nonlinear physical phenomena. In the recent decade, several methods for finding the exact solutions to nonlinear equations of mathematical physics have been proposed, such as trigonometric function series method [12], the modified mapping method and the extended mapping method [13] [14], homogeneous balance method [15], tanh function method [16], extended tanh function method [17], hyperbolic function method [18], rational expansion method [19], sine-cosine method [20], Jacobi elliptic function method [21], F-expansion method [22], and so on. Our goal is to present a method to computational study of solitary waves in nonlinear RLC transmission lines.…”

Section: Introductionmentioning

confidence: 99%

Propagation of Dark Solitary Waves in the Korteveg-Devries-Burgers Equation Describing the Nonlinear RLC Transmission

Yamigno¹

2014

JMP

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We investigate the propagation of dark solitons in a nonlinear dissipative electrical line. We show that the dynamics of the line is reduced to an expanded Korteweg-de Vries-Burgers (KdVB) equation. By applying the perturbation theory to the KdVB equation, we obtain soliton-like pulse solutions. The numerical simulations of the discrete equation are carried out and show the possibility of the founding solution to spread through the line. The effect of the dissipation through soliton is also shown. A chaotic-like behavior can take place in the system during the propagation of dark solitons through the line.

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New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water

Cited by 322 publications

References 11 publications

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

Some new solutions of the (1+1)-dimensional PDE via the improved (G′/G)-expansion method

Propagation of Dark Solitary Waves in the Korteveg-Devries-Burgers Equation Describing the Nonlinear RLC Transmission

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