Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach

Du, Zhijiang; Liu, Jiang; Ren, Yulin

doi:10.1016/j.jde.2020.09.009

Cited by 25 publications

(41 citation statements)

References 37 publications

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“…Based on the geometric singular perturbation approach to track invariant manifolds of corresponding ordinary differential equations (ODEs), Du and Qiao [39] explained existence of traveling wave solutions in a Belousov-Zhabotinskii system with delay by combing Fredholm orthogonality and asymptotic theory. Du et al [40] also confirmed the persistence of solitary wave solution of a generalized Keller-Segel system with small cell diffusion by Poincaré-Bendixson theorem. Clearly, the idea of detecting monotonicity of the ratio of Abelian integrals (MRAI) to prove existence of traveling wave solutions have been valid.…”

Section: Historymentioning

confidence: 85%

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: 85%

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

Zheng¹,

Xia²

2022

Preprint

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“…We mainly use geometric singular perturbation theory, the Fredholm theory and the linear chain trick. The corresponding convolution in equation ( 9) is described by (12), in which the kernel f : [0, +∞) → [0, +∞) satisfies the following normalization assumption,…”

Section: Solitary Waves For Generalized Kawahara Equation (9)mentioning

confidence: 99%

“…To deal with singular perturbations, an useful approach is the geometric singular perturbation theory [16], which can ensure the existence of the invariant manifold and the problem will be reduced to a regular perturbation system on the manifold. It has been successfully applied to analyze the perturbed Benjamin-Bona-Mahony equation [17,35], Camassa-Holm equation [11], Keller-Segel system [12], Belousov-Zhabotinskii system [13] and perturbed KdV equations [33] etc.…”

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confidence: 99%

The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation

Chen¹,

Du²,

Liu³

et al. 2022

DCDS-B

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In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.

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“…The Poincaré-Bendixson theorem is one of the most fundamental tools to capture the limit behaviors of orbits of flows and was applied to various phenomena (e.g. [5,10,17,19,[29][30][31][32]). In [6], Birkhoff introduced the concepts of ω-limit set and α-limit set of a point.…”

Section: Introductionmentioning

confidence: 99%

The $ω$-limit set of a flow with arbitrarily many singular points on a surface and surgeries to add singular points

Yokoyama¹

2022

Preprint

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The ω-limit set is one of the fundamental objects in dynamical systems. Using the ω-limit sets, the Poincaré-Bendixson theorem captures the limit behaviors of orbits of flows on surfaces. It was generalized in several ways and was applied to various phenomena. In this paper, we consider what kinds of the ω-limit sets do appear in the non-wandering flows on compact surfaces, and show that the ω-limit set of any non-closed orbit of a non-wandering flow with arbitrarily many singular points on a compact surface is either a subset of singular points or a locally dense Q-set. To show this, we demonstrate a similar result holds for locally dense orbits without non-wanderingness. Moreover, the ω-limit set of any non-closed orbits of a Hamiltonian flow with arbitrarily many singular points on a compact surface consists of singular points.

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Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach

Cited by 25 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

The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation

The $ω$-limit set of a flow with arbitrarily many singular points on a surface and surgeries to add singular points

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