New exact solutions for the generalized variable-coefficient Gardner equation with forcing term

Hong, Baojian; Seadawy, Aly R.

doi:10.1016/j.amc.2012.08.104

Cited by 22 publications

(21 citation statements)

References 24 publications

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“…Using the Bell polynomials, we have obtained Hirota Bilinear Form (16) of (2). By virtue of (16), two kinds of the multi-soliton solutions [i.e., (31) and (17)] and another kind of the analytic solutions [i.e., (37)] have been obtained, both with different nonlinear dispersion relations, as illustrated in Figures 1-7. For investigating the integrability of (2), we have obtained Lax Pairs (38) and BTs (44) for (2), with our integrability results as the multi-soliton solutions.…”

Section: Discussionmentioning

confidence: 99%

“…Higher-dimensional NLEEs are scientifically interesting: For example, some (3+1)-dimensional KdV-type equations can describe the dust-ion-acoustic waves in cosmic nonmagnetised dusty plasmas such as those in the supernova shells and Saturn's F-ring [28], and some (3+1)-dimensional NLEEs, with certain parameters, can reduce to the (2+1)-dimensional and (1+1)-dimensional NLEEs [29][30][31][32][33][34][35] where u is an analytic function depending on the scaled spatial coordinates (x, y, z) and temporal coordinate t. By virtue of the exp-function method [10] and bilinear method via the logarithm transformation [36], the soliton solutions of (2) have been discussed [10,36]. In the fluid and plasma dynamics, special cases of (2) are seen as follows: (a) When y = z, u(x, y, t) = (η, t) =  [x + χ(y), t] with as χ an analytic function of y, (2) degenerates into the KdV equation [37],…”

Section: Introductionmentioning

confidence: 99%

“…Three solitons in Case I of(31) and(17) with the parameters Φ(y) = Ψ(z) = 1, l 1 = 1.2, l 2 = 1.6, l 3 = 1.7, m 1 = 1.3, m 2 = 1.4, m 3 = 1.15, k 1 = 1.07, k 2 = 1.9, k 3 = 1.7, δ 1 = 1, δ 2 = 1.6, δ 3 = 1.5; and (a) of y = z = 1, (b) of z = t = 1, (c) of y = t = 1. Two solitons in Case II of (30) with the parameters sin Φ δ 1 = 0, δ 2 = 2, and (a) of y = z = 1, (b) of z = t = 1, (c) of y = t = 1.…”

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confidence: 99%

“…Three solitons in Case II of(31) and(17)with the parameters and (a) of y = z = 1. (b) of z = t = 1, (c) of y = t = 1.…”

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confidence: 99%

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On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

Zuo¹,

Gao²,

Yu³

et al. 2015

Zeitschrift Für Naturforschung A

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The Boiti-Leon-Manna-Pempinelli (BLMP) equation is seen as a model for the incompressible fluid. In this article, a (3+1)-dimensional BLMP equation is investigated. With the aid of the Bell polynomials, bilinear form of such an equation is obtained. By virtue of the bilinear form, two kinds of soliton solutions with different nonlinear dispersion relations and another kind of analytic solutions are derived. Lax pairs and Bäcklund transformations are also constructed. Soliton propagation and interaction are analysed: (i) solitions with different nonlinear dispersion relations have different velocities and backgrounds; (ii) for another kind of analytic solutions with different nonlinear dispersion relations, the periodic property is displayed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…Three solitons in Case II of(31) and(17)with the parameters and (a) of y = z = 1. (b) of z = t = 1, (c) of y = t = 1.…”

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confidence: 99%

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On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

Zuo¹,

Gao²,

Yu³

et al. 2015

Zeitschrift Für Naturforschung A

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“…We make the transformation By using the general mapping deformation method [10,40], we can obtain the following solutions of the corresponding undisturbed Eq. (70) when f ¼ 0.…”

Section: Application To the Generalized Perturbed Nls Equationmentioning

confidence: 99%

Homotopy Asymptotic Method and Its Application

Hong¹

2017

Recent Studies in Perturbation Theory

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As we all know, perturbation theory is closely related to methods used in the numerical analysis fields. In this chapter, we focus on introducing two homotopy asymptotic methods and their applications. In order to search for analytical approximate solutions of two types of typical nonlinear partial differential equations by using the famous homotopy analysis method (HAM) and the homotopy perturbation method (HPM), we consider these two systems including the generalized perturbed Kortewerg-de VriesBurgers equation and the generalized perturbed nonlinear Schrödinger equation (GPNLS). The approximate solution with arbitrary degree of accuracy for these two equations is researched, and the efficiency, accuracy and convergence of the approximate solution are also discussed.

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New exact solutions for the generalized variable-coefficient Gardner equation with forcing term

Cited by 22 publications

References 24 publications

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

Homotopy Asymptotic Method and Its Application

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