Classification of bounded travelling wave solutions for the Dullin–Gottwald–Holm equation

Silva, Priscila Leal da

doi:10.1016/j.jmaa.2018.10.086

Cited by 15 publications

(17 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us first recall that for α = β = γ = Γ = 0 and ǫ = 1, the classification of bounded traveling wave solutions was made in [41]. For β = γ = 0 corresponding to the Dullin-Gottwald-Holm equation, the analysis was recently considered by the first author in [21]. Therefore, in what follows, we will only consider (γ, β) = (0, 0).…”

Section: Classification Of Bounded Travelling Wave Solutionsmentioning

confidence: 99%

“…Our strategy to classify the bounded travelling waves for (1.0.4) is the following: firstly we use the first conservation law in Theorem 3.1 to obtain a quadrature to equation (1.0.4). Then we carry out a classification of the waves following the ideas of [41], see also [21].…”

Section: Classification Of Bounded Travelling Wave Solutionsmentioning

confidence: 99%

“…Before proceeding with the results, we shall consider some aspects of the theory that will be used here. In some parts of the text we follow closely the presentation in [21,41,43]. We begin with notation: z ↑ z 0 means that z < z 0 and z → z 0 , whereas z ↓ z 0 means z → z 0 , but z > z 0 .…”

Section: Preliminaries and Types Of Wavesmentioning

confidence: 99%

See 2 more Smart Citations

Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation

Silva

Freire

2019

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

In this paper we consider a five-parameter equation including the Camassa-Holm and the Dullin-Gottwald-Holm equations, among others. We prove the existence and uniqueness of solutions of the Cauchy problem using Kato's approach. Conservation laws of the equation, up to second order, are also investigated. From these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it. The quadrature obtained is of capital importance in a classification of bounded travelling wave solutions. We also find some explicit solutions, given in terms of elliptic integrals. Finally, we classify the members of the equation describing pseudo-spherical surfaces.

show abstract

Section: Classification Of Bounded Travelling Wave Solutionsmentioning

confidence: 99%

Section: Classification Of Bounded Travelling Wave Solutionsmentioning

confidence: 99%

See 1 more Smart Citation

Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation

Silva

Freire

2019

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…In Yu, 37 the dynamical behaviour of traveling wave solutions and its bifurcations were presented in different parameter regions. The classification of bounded travelling wave solutions were discussed in da Silva 38 …”

Section: Introductionmentioning

confidence: 99%

Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis

Bilal

Seadawy

Younis

et al. 2020

Math Methods in App Sciences

114

View full text Add to dashboard Cite

This article possesses modulation instability (MI) analysis and new exact wave solutions to unidirectional Dullin–Gottwald–Holm (DGH) system that describes the prorogation of waves in shallow water. The exact wave solutions in single and combined form like shock, singular, and shock‐singular are extracted by means of an innovative integration norm, namely, ()G′false/G2‐expansion scheme. The periodic and plane wave solutions are also emerged. The constraint conditions which ensure the existence of solutions are discussed as well. Moreover, the choice of suitable parameters gives the three‐dimensional and two‐dimensional sketches, and furthermore, their contour plots are also drawn.

show abstract

“…It reduces to the CH equation if λ = β = γ = Γ = 0, whereas the Dullin-Gottwald-Holm equation [14,19] is recovered when λ = β = γ = 0 and αΓ = 0. If β = γ = Γ = 0 and λ > 0 we have the weakly dissipative CH equation [42,43], whereas if λ > 0, αΓ = 0 and β = γ = 0 we have the weakly dissipative DGH equation [35,36,44].…”

Section: Introductionmentioning

confidence: 99%

Wave breaking for shallow water models with time decaying solutions

Freire

2019

Preprint

View full text Add to dashboard Cite

A family of Camassa-Holm type equations with a linear term and cubic and quartic nonlinearities is considered. Local well-posedness results are established via Kato's approach. Conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is time-dependent. If such coefficient is positive, then the energy functional is a monotonically decreasing function of time, bounded from above by the Sobolev norm of the initial data, and all solutions of the equation are dissipative. Sufficient conditions for the global existence of solutions are described. The existence of wave breaking phenomena is also investigated and necessary conditions for its existence are obtained. In our framework the wave breaking is guaranteed, among other conditions, when the coefficient of the linear term is sufficiently small, which allows us to interpret the equation as a linear perturbation of some recent Camassa-Holm type equations considered in the literature. 2010 AMS Mathematics Classification numbers: 35A01, 35L65, 37K05.

show abstract

Classification of bounded travelling wave solutions for the Dullin–Gottwald–Holm equation

Cited by 15 publications

References 23 publications

Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation

Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation

Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis

Wave breaking for shallow water models with time decaying solutions

Contact Info

Product

Resources

About