Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers–KdV equation

Xu, Ying; Du, Zhijiang; Lei, Wei

doi:10.1007/s11071-015-2309-5

Cited by 26 publications

(11 citation statements)

References 40 publications

(68 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The boundary layer series Π (+) (ξ + , t, ) is constructed with the similar method of the boundary layer series Π (−) (ξ − , t, ). Its terms admit an estimate of the form (35) with ξ − replaced by ξ + . 4.…”

Section: Q (∓)mentioning

confidence: 99%

“…In the model for such phenomena, it is stated that the dependence of the physical variables on the small parameter equal to ratio of the transition layer width to the width of the domain under study. In recent years, the geometric singular perturbation theory is applied to study the biological model such as predator-prey model and disease model [5,3,34,35,28,4]. In [5], Hek explained and explored the three Fenichel's main theorems and its significance and applications in biological practice with many examples.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Wu¹,

Ni²

2021

DCDS-S

View full text Add to dashboard Cite

In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

show abstract

Section: Q (∓)mentioning

confidence: 99%

mentioning

confidence: 99%

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Wu¹,

Ni²

2021

DCDS-S

View full text Add to dashboard Cite

show abstract

“…Tang et al [20] investigated the persistence of the solitary wave solution for singularly perturbed Gardner equation. Xu et al [21] established the existence of wave fronts for a generalized Burgers-KdV equation with convolution kernel. Du et al [22,23] studied the existence of solitary wave solutions for delayed CamassaHolm equation and also considered the existence of wave fronts for nonlinear Belousov-Zhabotinskii system with delay.…”

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

View full text Add to dashboard Cite

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.

show abstract

“…Lodhi and Mishra [12] discussed second order singularly perturbed nonlinear boundary value problems by using the quintic B-spline method. Recently, the geometric singular perturbation theory has also received a great deal of interests in studying the Burgers-KdV equation [13], the vector-disease model [14], the perturbed BBM equation [15], the perturbed Camassa-Holm equation [16] and the perturbed shallow water wave model [17] etc.…”

Section: Introductionmentioning

confidence: 99%

The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

2020

View full text Add to dashboard Cite

In this paper, we discuss third-order full nonlinear singularly perturbed vector boundary value problems. We first present the existence of solutions for the nonlinear vector boundary value problems without perturbation by using the upper and lower solutions method and topological degree theory. Then the existence, uniqueness and asymptotic estimates of solutions for the singularly perturbed vector boundary value problems are established by constructing appropriate a lower solution-upper solution pair, as well as analysis technique. Some known results are extended.

show abstract

Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers–KdV equation

Cited by 26 publications

References 40 publications

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

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

The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

Contact Info

Product

Resources

About