Painlevé analysis, complete Lie group classifications and exact solutions to the time-dependent coefficients Gardner types of equations

Liu, Hanze; Li, Jibin

doi:10.1007/s11071-014-1885-0

Cited by 15 publications

(8 citation statements)

References 27 publications

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“…Many powerful methods have been presented for finding the traveling wave solutions of nonlinear partial differential equations, such as the Bäcklund transformation [10], Darboux transformation [11], inverse scattering method [12], Hirota bilinear method [13], Lie group analysis method [14][15][16], tanh method [17], ansatz method [18,19], bifurcation theory of dynamical system [20,21], exp-function method [22,23], symbolic computation method [24][25][26], and other methods [27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

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Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

Meng

2016

Nonlinear Dyn

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A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and antikink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

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Section: Introductionmentioning

confidence: 99%

“…Parameters: a h = 0. b h Limiting precess of u15 (ξ ) tends to u 13 (ξ ) when h → 0. Parameters: a h = −3.0.…”

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confidence: 99%

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

Meng

2016

Nonlinear Dyn

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“…Because for understanding the nonlinear phenomena, which are usually described by partial differential equations, the study of the exact solutions is essential. Many powerful methods have been presented for finding the traveling wave solutions of nonlinear partial differential equations, such as the Bäcklund transformation [1], Darboux transformation [2], inverse scattering method [3], Hirota bilinear method [4], Lie group analysis method [5], transformed rational function method [6], extended transformed rational function method [7], exp-function method [8], multiple exp-function algorithm [9], and symbolic computation method [10].…”

Section: Introductionmentioning

confidence: 99%

Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

Meng

2016

Discrete Dynamics in Nature and Society

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The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

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“…For this reason, the scope of this paper is to explore the relationship between the existence of the infinitely many solitary waves and the wave speed for Eq. (1.5) by adopting the bifurcation method of dynamical systems [6,7,[16][17][18][19][20][21][22][23]. We list the other approaches for solving the solitary waves for comparison [5,[8][9][10]24,25].…”

Section: Introductionmentioning

confidence: 99%

Infinitely many solitary waves of an integrable equation with singularity

Pan

Liu

2015

Nonlinear Dyn

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Little seems to be known about the solitary waves and their properties of the completely integrable equations with singularity. This paper addresses the solitary waves of an integrable equation based on the bifurcation method of dynamical systems. We highlight two interesting results on the solitary waves. First, for arbitrary wave speed, there do exist infinitely many solitary waves in the integrable equation, which are classified by their expressions and forms of motion. Second, we find a family of solitary waves whose profiles seem like tree stumps.

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Painlevé analysis, complete Lie group classifications and exact solutions to the time-dependent coefficients Gardner types of equations

Cited by 15 publications

References 27 publications

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

Infinitely many solitary waves of an integrable equation with singularity

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