Existence of Solitary Waves and Periodic Waves to a Perturbed Generalized KDV Equation

Yan, Weifang; Liu, Zhengrong; Liang, Yuelong

doi:10.3846/13926292.2014.960016

Cited by 41 publications

(27 citation statements)

References 19 publications

(23 reference statements)

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“…After the works of Derks and Gils [9] and Ogawa [36], in 2014 Yan et al [47] investigated the perturbed generalized KdV equation,…”

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

“…Yan et al [47] proved that there exists one periodic wave by choosing some wave speed c for sufficiently small > 0. However, the uniqueness of the periodic wave is still open.…”

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

“…and established the existence of solitary waves and uniqueness of periodic waves. Both of the works [47] and [7] studied the perturbation problems restricted on manifolds, by using geometric singular perturbation theory. In [7], the authors applied Picard-Focus equations to determine the existence of periodic waves, and developed a good approach to prove that the dominating factor of the Melnikov function is monotonic, see Lemma 4.10 in [7].…”

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

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Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Sun¹,

Yu²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the coexistence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

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“…After the works of Derks and Gils [9] and Ogawa [36], in 2014 Yan et al [47] investigated the perturbed generalized KdV equation,…”

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

“…Yan et al [47] proved that there exists one periodic wave by choosing some wave speed c for sufficiently small > 0. However, the uniqueness of the periodic wave is still open.…”

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

See 1 more Smart Citation

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Sun¹,

Yu²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…The traveling wave solutions, especially the solitary wave solutions of these equations have attracted extensive attentions and some effective techniques or methods have been proposed, among which the dynamical system method [10][11][12][13][14] are well applied to investigate the traveling wave solutions of various nonlinear wave equations. In particularly, it has been shown that equation (4) (or(5)) with n > 1 admits a kind of solitary waves with compact support and therefore are named as compactons [6,8,9].…”

Section: Introductionmentioning

confidence: 99%

“…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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A Hybridized Discontinuous Galerkin Method for the Nonlinear Korteweg–de Vries Equation

et al. 2015

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Existence of Solitary Waves and Periodic Waves to a Perturbed Generalized KDV Equation

Cited by 41 publications

References 19 publications

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

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

A Hybridized Discontinuous Galerkin Method for the Nonlinear Korteweg–de Vries Equation

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