On the Cauchy problem of generalized Fokas–Olver–Resenau–Qiao equation

Yang, Meiling; Li, Yongsheng; Zhao, Yongye

doi:10.1080/00036811.2017.1359565

Cited by 24 publications

(16 citation statements)

References 33 publications

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“…Lemma 2.1. [41] Let u 0 (x) = u(0, x) ∈ H s (R) with s > 5/2. Then there exists a time T > 0 such that the Cauchy problem of (1.1) has a unique strong solution u(t, x) ∈ C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)) and the map u 0 → u is continuous from a neighborhood of…”

Section: Preliminariesmentioning

confidence: 99%

Stability of peakons for the generalized modified Camassa–Holm equation

Guo

Liu

et al. 2019

Journal of Differential Equations

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In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function h(t, x) and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Comm. Math. Phys., 322:967-997, 2013) successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics.Recently, the great interest in the CH equation has inspired the search for various CH-type equations with cubic or higher-order nonlinearities. One of the most concerned is the following modified CH (mCH) equationwas derived by Fuchssteiner [22] and Olver and Rosenau [32] by employing the tri-Hamiltonian duality approach to the bi-Hamiltonian representation of the modified KdV equation. Subsequently, the integrability and structure of solutions to the mCH equation was discussed by Qiao [33]. The mCH equation is also called FORQ equation in some literature [26, 41]. The mCH equation is completely integrable [32]. It has a bi-Hamiltonian structure and also admits a Lax pair [35], and hence may be solved by the inverse scattering transform method. Compared with the CH equation, the mCH equation admits not only peakons, but also possesses cusp solitons (cuspons) and weak kink solutions (u, u x , u t are continuous, but u xx has a jump at its peak point) [35,40]. It has also significant differences from the CH equation on the dynamics of the multi-peakons and peakon-kink solutions [24,34,40]. Fu et al.,[21] studied the Cauchy problem of the mCH equation in Besov spaces and the blow-up scenario. The nonuniform dependence on the initial data was established in [26]. Gui et al.,[24] considered the formulation of singularities of solutions and showed that some solutions with certain initial data would blow up in finite time. Then the blow-up phenomena were systematically investigated in [7,31]. The mCH equation admits single peakon of the form [24] u(t, x) = ϕ c (x − c t) = 3 c 2 e −|x−c t| , c > 0. Later, inspired by [16] and [18], the orbital stability of single peakon and the train of peakons for the mCH equation were proved in [36] and [30], respectively.

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Section: Preliminariesmentioning

confidence: 99%

Stability of peakons for the generalized modified Camassa–Holm equation

Guo

Liu

et al. 2019

Journal of Differential Equations

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“…has been studied for a long time and achieved so many results, such as the solution of Cauchy initial value problem [1], Holder continuous [2], the algebro-geometric solutions [3], the Cauchy problem of the generalized equation [4], and the nonuniqueness for the equation [5]. A lot of solving methods for the nonlinear partial differential equation are discussed to be applied in engineering and practice areas, such as the sine-Gordon expansion method [6] and the travelling wave method and its conservation laws [7], and so many examples are in this regard.…”

Section: Introductionmentioning

confidence: 99%

Traveling Wave Solution of the Olver–Rosenau Equation Solved by Dynamics System

Xiong

Chen

Yang

2020

Mathematical Problems in Engineering

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Olver–Rosenau equations presented by Olver and Rosenau can be rewritten to the dynamic system by the wave transformation. The system is a Hamiltonian system with the first integral, and its phase-space and equilibrium point analysis are given in different parameter spaces in detail. On this basis, we can derive various solutions of the original equation relating these orbits in different phase-space planes, and the theoretical basis of the numerical solution is provided for engineering application and production practice.

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“…The following local well-posedness result and properties for strong solutions on unit circle (up to a slight modification) were established in [26] and [56]. Proposition 1.…”

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confidence: 96%

Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation

Moon¹

2021

DCDS-S

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<p style='text-indent:20px;'>This paper is devoted to studying the dynamical stability of periodic peaked solitary waves for the generalized modified Camassa-Holm equation. The equation is a generalization of the modified Camassa-Holm equation and it possesses the Hamiltonian structure shared by the modified Camassa-Holm equation. The equation admits the periodic peakons. It is shown that the periodic peakons are dynamically stable under small perturbations in the energy space.</p>

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On the Cauchy problem of generalized Fokas–Olver–Resenau–Qiao equation

Cited by 24 publications

References 33 publications

Stability of peakons for the generalized modified Camassa–Holm equation

Stability of peakons for the generalized modified Camassa–Holm equation

Traveling Wave Solution of the Olver–Rosenau Equation Solved by Dynamics System

Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation

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