Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method

Zhuang, Kaige; Du, Zhijiang; Lin, Xin

doi:10.1007/s11071-015-1894-7

Cited by 22 publications

(11 citation statements)

References 34 publications

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“…Clearly, the perturbed KdV equation ( 6) is a special case l = 1 and m = 2 of equation ( 7), so it is a generalization of the perturbed KdV equation. Guo and Zhao [17] have examined the existence of periodic waves for equation (7) with l = 3 and m = 5; The particular case that (7) with l = 1 and arbitrary m ∈ Z + , also named as singularly perturbed higher-order KdV equation, has been investigated in [12,18] via geometric singular perturbation theory.…”

Section: Introductionmentioning

confidence: 99%

New Type of Solitary Wave Solution With the Coexisting Peak and Valley for a Perturbed Wave Equation

Zhang

Wang

Shchepakina

et al. 2021

Preprint

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The perturbed mK(3,1) equation is restudied to further explore the dynamics of solitary wave solutions by combining the geometric singular perturbation theorem and bifurcation analysis in this paper. Besides the solitary waves presented in literature, we show that this equation possesses a family of solitary waves which decay to some constants determined by their wave speeds and a parameter. It is shown that a portion of the solitary wave solutions to the mK(3,1) equation will persist under small perturbations and the wave speed selection principle is presented as well. In addition to the solitary waves, each of which has only one peak or valley and approximates to a solitary wave of the unperturbed equation as the perturbation parameter tends to zero, we theoretically prove the existence of a new type of solitary waves with the coexisting peak and valley. The numerical simulations are carried out, and the results are in complete agreement with our theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

New Type of Solitary Wave Solution With the Coexisting Peak and Valley for a Perturbed Wave Equation

Zhang

Wang

Shchepakina

et al. 2021

Preprint

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“…In recent years, nonlinear behaviors of the soliton equations under the external perturbations have been paid much attentions, since there often exist various perturbations in real physical systems [1][2][3][4][5][6][7][8][9][10][11][12]. For instance, the perturbations arising in Korteweg-de Vries (KdV) equation can be used to describe the resonant forcing in a tank of finite length [1], traveling steady pressure distribution on the water of finite depth [2] and solitons generated by moving pressure disturbances [3]; the perturbed sine-Gordon equation was used to model the propagations of nonlinear waves through the quasi-periodic or chaotic media [5].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

Wang

2016

Nonlinear Dyn

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The nonlinear dynamics of a generalized higher-order nonlinear Schrödinger (HNLS) equation with a periodic external perturbation is investigated numerically. Via the phase plane analysis, we find that both the homoclinic orbits and heteroclinic orbits can exist for the unperturbed HNLS equation under certain conditions, which respectively corresponds to the bellshaped and kink-shaped soliton solutions. Moreover, under the effect of the periodic external perturbation, the quasi-periodic bifurcations arise and can evolve into the chaos. The dynamical responses of the perturbed system varying with the perturbation strength and two types of chaotic attractors are discussed to show the existence of the chaotic motions. Via the feedback control methods, such chaotic motions are found to be controlled effectively and finally evolve into the stable quasi-periodic orbits. All the results are helpful to understand the dynamical properties of the nonlinear system.

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“…Geometric singular perturbation theory has been used by many researchers to obtain the existence of traveling waves and solitary waves [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. For example, Xu and Du [2,21] used geometric singular perturbation theory, Fredholm theory, and the linear chain trick to investigate the existence of traveling wave and solitary wave solutions for some generalized KdV equations. In [19,20,22], the authors discussed the predator-prey systems and population genetics model by using geometric singular perturbation method combing with the Hopf bifurcation theorem, the improved shooting method.…”

Section: Introductionmentioning

confidence: 99%

“…It uses invariant manifolds in phase space to understand the global structure of the phase space or to construct orbits with desired properties . Geometric singular perturbation theory has been used by many researchers to obtain the existence of traveling waves and solitary waves . For example, Xu and Du used geometric singular perturbation theory, Fredholm theory, and the linear chain trick to investigate the existence of traveling wave and solitary wave solutions for some generalized KdV equations.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves in a generalized nonlinear dispersive–dissipative equation

Shang

2015

Math Methods in App Sciences

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In this paper, we consider the existence of traveling waves in a generalized nonlinear dispersive-dissipative equation, which is found in many areas of application including waves in a thermoconvective liquid layer and nonlinear electromagnetic waves. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory and invariant manifold theory, Fredholm theory, and the linear chain trick, we construct a locally invariant manifold for the associated traveling wave equation and use this invariant manifold to obtain the traveling waves for the nonlinear dispersive-dissipative equation.

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Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method

Cited by 22 publications

References 34 publications

New Type of Solitary Wave Solution With the Coexisting Peak and Valley for a Perturbed Wave Equation

New Type of Solitary Wave Solution With the Coexisting Peak and Valley for a Perturbed Wave Equation

Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

Traveling waves in a generalized nonlinear dispersive–dissipative equation

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