An equation unifying both Camassa-Holm and Novikov equations

Silva, Priscila Leal da; Freire, Igor Leite

doi:10.3934/proc.2015.0304

Cited by 6 publications

(4 citation statements)

References 27 publications

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“…Since its derivation, other equations sharing common properties with the CH equation have been discovered and investigated from both mathematical or physical motivations [3,19,20,23,24,30,31,35,50,66], which are also subject of interest for a wide spectra of researchers and from different frames, see [3,5,19-22, 32-34, 45, 49, 50, 56, 57, 63] and references therein. Moreover, modifications, extensions and generalizations of these equations have also been intensely studied, see [3,20,32,45,49,50,63] and references thereof.…”

Section: Introductionmentioning

confidence: 99%

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

Silva

Freire

2019

Journal of Differential Equations

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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.

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Section: Introductionmentioning

confidence: 99%

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

Silva

Freire

2019

Journal of Differential Equations

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“…The 4-parameter equation (1.1) studied in [1,15,16], is a generalization of the Camassa-Holm equation [6]…”

Section: Introductionmentioning

confidence: 99%

Local well-posedness and global analyticity for solutions of a generalized 0-equation

Silva¹

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this work we study the Cauchy problem in Gevrey spaces for a generalized class of equations that contains the case $b=0$ of the $b$ -equation. For the generalized equation, we prove that it is locally well-posed for initial data in Gevrey spaces. Moreover, as we move to global well-posedness, we show that for a particular choice of the parameter in the equation the local solution is global analytic in both time and spatial variables.

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“…In [21], under the assumption (14), the convergence of the solution of (1) to the unique entropy solution of (13) is proven. In order to do this, the author used kinetic methods, which were introduced in [22].…”

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confidence: 98%

“…Introduction. Consider the Camassa Holm type equations (see [14]) ∂ t u − α∂ 3 txx u + ∂ x f (u) = αnu m ∂ x u∂ 2 xx u + αu m+1 ∂ 3 xxx u, n, m ∈ N.…”

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confidence: 99%

A note on the convergence of the solution of the Novikov equation

Coclite¹,

Ruvo²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We consider the Novikov and Camass-Holm equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.

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An equation unifying both Camassa-Holm and Novikov equations

Cited by 6 publications

References 27 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

Local well-posedness and global analyticity for solutions of a generalized 0-equation

A note on the convergence of the solution of the Novikov equation

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