On the wave length of smooth periodic traveling waves of the Camassa–Holm equation

Geyer, Anna; Villadelprat, Jordi

doi:10.1016/j.jde.2015.03.027

Cited by 31 publications

(23 citation statements)

References 61 publications

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“…This result was obtained in Theorem 2.5 of Ref. 18, where the second-order equation ( 6) with the first-order invariant (8) was reformulated into the system…”

mentioning

confidence: 77%

“…In comparison with Theorem 1, it follows from the results in Ref. 18 that the period of the periodic solutions of the system ( 6)-( 8)…”

Section: Introductionmentioning

confidence: 86%

“…Remark 3. In the context of Theorem 2, we show numerically that the stability criterion (18) is satisfied for every periodic solution of Theorem 1, see Figure 7.…”

Section: Introductionmentioning

confidence: 93%

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Stability of smooth periodic travelling waves in the Camassa–Holm equation

et al. 2021

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We solve the open problem of spectral stability of smooth periodic waves in the Camassa-Holm equation. The key to obtaining this result is that the periodic waves of the Camassa-Holm equation can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg-de Vries equation. The standard formulation has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We prove that the nonstandard formulation has the advantage that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.

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“…This result was obtained in Theorem 2.5 of Ref. 18, where the second-order equation ( 6) with the first-order invariant (8) was reformulated into the system…”

mentioning

confidence: 77%

“…In comparison with Theorem 1, it follows from the results in Ref. 18 that the period of the periodic solutions of the system ( 6)-( 8)…”

Section: Introductionmentioning

confidence: 86%

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Stability of smooth periodic travelling waves in the Camassa–Holm equation

et al. 2021

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“…Remark 2.4. It follows from the recent work in [17,18] that the amplitude-to-period map for the differential equations (2.2) and (2.3) is strictly monotonically increasing. With the scaling transformation, this result can be used to prove monotonicity of the parameters γ and I in terms of the wave amplitude a, which are expanded asymptotically as a → 0 by (2.5) and (2.6).…”

Section: Travelling Wavesmentioning

confidence: 98%

Orbital stability of periodic waves in the class of reduced Ostrovsky equations

Johnson

Pelinovsky

2016

Journal of Differential Equations

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Periodic travelling waves are considered in the class of reduced Ostrovsky equations that describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky equations with either quadratic or cubic nonlinearities can be transformed to integrable equations of the Klein-Gordon type by means of a change of coordinates. By using the conserved momentum and energy as well as an additional conserved quantity due to integrability, we prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. The proof is based on construction of a Lyapunov functional, which is convex at the periodic wave and is conserved in the time evolution. We also show numerically that convexity of the Lyapunov functional holds for periodic waves of arbitrary amplitudes.

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“…However, the weak Marangoni effect may destabilize the waves [21], and different perturbations may generate different dynamics of systems, leading to, for example, broking the periodic traveling waves, changing its stability and yielding quasi-periodic motions on invariant tori, etc. One efficient way to deal with such problems is to apply bifurcation techniques from the view point of dynamical systems by taking the weak external effects as perturbations, and many good results have been obtained for certain nonlinear wave problems, see [48,17,22,13].…”

Section: Xianbo Sun and Pei Yumentioning

confidence: 99%

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Sun¹,

Yu²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the coexistence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

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On the wave length of smooth periodic traveling waves of the Camassa–Holm equation

Cited by 31 publications

References 61 publications

Stability of smooth periodic travelling waves in the Camassa–Holm equation

Stability of smooth periodic travelling waves in the Camassa–Holm equation

Orbital stability of periodic waves in the class of reduced Ostrovsky equations

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

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