Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation

Du, Zhijiang; Ji, Li

doi:10.1016/j.jde.2021.10.033

Cited by 23 publications

(9 citation statements)

References 37 publications

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“…There exists an extensive literature on the use of geometric singular perturbation theory [22,23] to deal with these issues including KdV equation [24,25], generalized KdV equation [26][27][28], delayed CH equation [29], generalized CH equation [30], CH Kuramoto-Sivashinsky equation [31], perturbed BBM equation [32], generalized BBM equation [33],…”

Section: Historymentioning

confidence: 99%

Geometric singular perturbation analysis to a perturbed (1 + 1)-dimensional dispersive long wave equation

Zheng¹,

Xia²

2022

Preprint

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The existence of the solitary wave and the nonexistence of kink (anti-kink) wave solutions are studied for a perturbed (1 + 1)-dimensional dispersive long wave equation. The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and bifurcation analysis. The results show that the solitary wave solution with a suitable wave speed c and parameter κ exists under the small singular perturbation. Interestingly, unlike solitary wave solutions, the kink (anti-kink) wave solution doesn't persist because the corresponding Melnikov function has no zeros. Further, numerical simulations are utilized to verify the correctness of our analytical results.

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Section: Historymentioning

confidence: 99%

Geometric singular perturbation analysis to a perturbed (1 + 1)-dimensional dispersive long wave equation

Zheng¹,

Xia²

2022

Preprint

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“…Note that, for the reduced system (15), which has a line of equilibrium given by Remark 1 This key information can not be drawn from solely investigation of system (15) because of the o( ) term, but from the special structure of the KdV equation. It is this property that makes possible another application of the method of singular perturbation.…”

Section: Following Normalization Assumptionmentioning

confidence: 99%

“…The perturbation terms, the backward diffusion u xx and dissipation u xxxx , are usually called the KS perturbation. In recent years the problem of persistence of solitary waves in shallow water wave equations with the KS perturbation has received much attention [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

2022

J Nonlinear Math Phys

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In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.

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“…In the case, the singular perturbed system can be reduced to a regular perturbed system on the invariant manifold, and the existence of invariant manifolds can also be ensured. Notably, geometric singular perturbation theory has a successfully application in some aspects, such as Camassa-Holm equations [12,13], Belousov-Zhabotinskii systems [15], FitzHugh-Nagumo equation [36] and so on. Nonetheless, it has been applied less frequently to address the Keller-Segel system.…”

Section: Introductionmentioning

confidence: 99%

Persistence of Traveling Waves to the Time Fractional Keller-Segel System With a Small Parameter

Chen¹,

Alsaedi²,

Stamova

2023

Preprint

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This paper aims to investigate the time fractional Keller-Segel system with a small parameter. After the fractional order traveling wave transformation, the heteroclinic orbit to the degenerate time fractional Keller-Segel system is demonstrated through the method of constructing a suitable invariant region. Moreover, the persistence of traveling waves in the system with a small parameter can be further illustrated. The results are mainly reliance on the application of geometric singular perturbation theory and Fredholm theorem, which are fundamental theoretical frameworks for dealing with problems of complexity and high dimensionality. Eventually, the asymptotic behavior is depicted by the asymptotic theory to illustrate the rate of decay for traveling waves.

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Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation

Cited by 23 publications

References 37 publications

Geometric singular perturbation analysis to a perturbed (1 + 1)-dimensional dispersive long wave equation

Geometric singular perturbation analysis to a perturbed (1 + 1)-dimensional dispersive long wave equation

Solitary Waves of the Perturbed KdV Equation with Nonlocal Effects

Persistence of Traveling Waves to the Time Fractional Keller-Segel System With a Small Parameter

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