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

Silva, Priscila Leal da; Freire, Igor Leite

doi:10.1016/j.jde.2019.05.033

Cited by 35 publications

(52 citation statements)

References 61 publications

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“…Our main interest in this paper is to go further with the investigation initiated in Ref. 34 with respect to the equation

\begin{matrix} true \\ right m_{t} + u m_{x} + 2 u_{x} m & left = α u_{x} + β u^{2} u_{x} + γ u^{3} u_{x} + normalΓ u_{x x x}, \end{matrix}

where

m = u - u_{x x}

α, β, γ

, and

normalΓ

are real arbitrary constants. Equation () clearly includes the CH and DGH equations as particular cases, and a more recent and also physically relevant member of () is

m_{t} + u m_{x} + 2 u_{x} m + c u_{x} - \frac{β_{0}}{β} u_{x x x} + \frac{ω_{1}}{α^{2}} u^{2} u_{x} + \frac{ω_{2}}{α^{3}} u^{3} u_{x} = 0,

with

\begin{matrix} true \\ right c & left = \sqrt{1 + Ω^{2}} - normalΩ, α = \frac{c^{2}}{1 + c^{2}}, β_{0} = \frac{c false(c^{4} + 6 c^{2} - 1 false)}{6 {false(c^{2} + 1 false)}^{2}}, β = \frac{3 c^{4} + 8 c^{2} - 1}{6 {false(c^{2} + 1 false)}^{2}}, \\ right ω_{1} & left = - 3 c (c^{2} - 1) (c^{...} \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation

Silva

Freire

2020

Stud Appl Math

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Recent generalizations of the Camassa-Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness in Sobolev spaces for the family under consideration. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Based on the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117-139 (2006)] regarding bi-Hamiltonian deformations, we introduce the notion of quasi-integrability and prove that there exists a unique bi-Hamiltonian structure for the equation only when it is reduced to the Dullin-Gotwald-Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis effects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin-Gotwald-Holm equation describes pseudo-spherical surfaces.

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“…Our main interest in this paper is to go further with the investigation initiated in Ref. 34 with respect to the equation

\begin{matrix} true \\ right m_{t} + u m_{x} + 2 u_{x} m & left = α u_{x} + β u^{2} u_{x} + γ u^{3} u_{x} + normalΓ u_{x x x}, \end{matrix}

where

m = u - u_{x x}

α, β, γ

, and

normalΓ

are real arbitrary constants. Equation () clearly includes the CH and DGH equations as particular cases, and a more recent and also physically relevant member of () is

m_{t} + u m_{x} + 2 u_{x} m + c u_{x} - \frac{β_{0}}{β} u_{x x x} + \frac{ω_{1}}{α^{2}} u^{2} u_{x} + \frac{ω_{2}}{α^{3}} u^{3} u_{x} = 0,

with

\begin{matrix} true \\ right c & left = \sqrt{1 + Ω^{2}} - normalΩ, α = \frac{c^{2}}{1 + c^{2}}, β_{0} = \frac{c false(c^{4} + 6 c^{2} - 1 false)}{6 {false(c^{2} + 1 false)}^{2}}, β = \frac{3 c^{4} + 8 c^{2} - 1}{6 {false(c^{2} + 1 false)}^{2}}, \\ right ω_{1} & left = - 3 c (c^{2} - 1) (c^{...} \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation

Silva

Freire

2020

Stud Appl Math

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“…Se (C 0 , C 1 )é um vetor conservado para uma equação do tipo (1) A equação (4) possui multiplicadores até ordem 2 e, consequentemente, leis de conservação até este tipo de ordem, se, e somente se, λ = 0. Se λ = 0, então as leis de conservação e suas respectivas quantidades conservadas são aquelas obtidas em [4]. Se λ = 0, então as quantidades conservadas vão a 0 quando t → ∞.…”

Section: Simetrias De Lie E Leis De Conservaçãounclassified

“…Estes são alguns aspectos, talvez os primeiros, que explicam as extraordinárias propriedades exibidas pela equação. Nas referências [4,6] pode-se encontrar uma série de outras 2 propriedades interessantes desta equação, bem como uma extensa lista de referências a trabalhos tendo como foco principal (1) ou equações dela obtidas.…”

Section: Introductionunclassified

“…Embora a equação (2) seja fisicamente relevante, parte das propriedades matemáticas elencadas no começo da seção são perdidas, dentre elas, a existência de solitons fracos quando (β, γ) = (0, 0). Por outro lado, os resultados em [3,4,7,8,15] são suficientes para mostrar a relevância matemática de (2), o qual ainda exibe uma série de propriedades bastante interessantes eé fisicamente relevante.…”

Section: Introductionunclassified

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Simetrias de Lie de Equações do Tipo Camassa-Holm Com Perda de Energia

Freire¹

2020

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…The behaviors of solutions to the CH equation with dissipative term and dispersion term are studied in [25]. The local well-posedness for the Cauchy problem of the CH type equations [6,15,20,26,[28][29][30][31], asymptotic stability [17,22], solitons solutions [14], and regularity of conservative solutions [18] are considered. The readers may refer to [8-10, 18, 20-22] Molinet [23] considers the peakon solutions of the DP equation.…”

Section: (11)mentioning

confidence: 99%

Well-posedness and behaviors of solutions to an integrable evolution equation

Ming

Lai

2020

Bound Value Probl

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This work is devoted to investigating the local well-posedness for an integrable evolution equation and behaviors of its solutions, which possess blow-up criteria and persistence property. The existence and uniqueness of analytic solutions with analytic initial values are established. The solutions are analytic for both variables, globally in space and locally in time. The effects of coefficients λ and β on the solutions are given.

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

Cited by 35 publications

References 61 publications

Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation

Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation

Simetrias de Lie de Equações do Tipo Camassa-Holm Com Perda de Energia

Well-posedness and behaviors of solutions to an integrable evolution equation

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