On the Cauchy problem for a generalized Camassa-Holm equation
with both quadratic and cubic nonlinearity

Liu, Xingxing; Zhang, Qiao; Yin, Zhaoyang

doi:10.3934/cpaa.2014.13.1283

Cited by 9 publications

(5 citation statements)

References 35 publications

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“…where a = (−3k 3 ± 9k 2 3 + 24ck 1 )/(4k 1 ) [66,76]; the mCH-Novikov equation for k 1 k 2 = 0, k 3 = 0, where a = ± 3c/(2k 1 + 3k 2 ) [70] (Notice that we here correct it); or the Novikov-CH equation for k…”

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities

Qina¹,

Yan²,

Guo³

2021

Preprint

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This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation ((alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory.Then, the H ölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.

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Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities

Qina¹,

Yan²,

Guo³

2021

Preprint

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show abstract

“…(1.1), its biHamiltonian structure, and peakons, weak kinks and kink-peakon interactional solutions have shown in [36]. Recently, we [31] have studied the non-periodic Cauchy problem for Eq. (1.1) with initial data u 0 ∈ B s p,r , for 1 ≤ p, r ≤ ∞, s > max{2 + 1 p , 5 2 , 3 − 1 p }.…”

Section: Introductionmentioning

confidence: 99%

“…The method used here is different from our approach to the non-periodic case in [31]. Inspired by [24], we also transform Eq.…”

Section: Introductionmentioning

confidence: 99%

The periodic Cauchy problem for a combined CH–mCH integrable equation

Liu

2016

Nonlinear Analysis: Theory, Methods & Applications

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“…One can refer to [39,45] for the study of its Lax pair, peakons, weak kinks, kinkpeakon interaction, and local well-posedness. As m 1 = n 2 , m 2 = n 1 , making the change of variable…”

Section: Introduction Recently LI Liu and Popowiczmentioning

confidence: 99%