On the Cauchy problem for a modified Camassa–Holm equation

Luo, Zhaonan; Zhang, Qiao; Yin, Zhaoyang

doi:10.1007/s00605-020-01426-3

Cited by 8 publications

(4 citation statements)

References 25 publications

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“…Several groups have studied the mCH equation from various aspects such as the well-posedness, regularization, Cauchy problem, Riemann-Hilbert problem, long-time asymptotics, and Liouville correspondence with the modified Korteweg-de Vries (KdV) equation. [7][8][9][10][11][12][13][14][15] The smooth soliton solutions were constructed by different approaches such as the Hirota's bilinear method 16,17 and Darboux transformation/Bäcklund transformation method. [18][19][20] Recently, Chang et al considered the Lax integrability and the conservative peakon solutions in a series of work [21][22][23] while Gao et al studied the patched peakon weak solution.…”

Section: Introductionmentioning

confidence: 99%

“…The mCH equation () has attracted considerable attention in the past two decades due to its rich mathematical structure and solutions. Several groups have studied the mCH equation from various aspects such as the well‐posedness, regularization, Cauchy problem, Riemann–Hilbert problem, long‐time asymptotics, and Liouville correspondence with the modified Korteweg‐de Vries (KdV) equation 7‐15 . The smooth soliton solutions were constructed by different approaches such as the Hirota's bilinear method 16,17 and Darboux transformation/Bäcklund transformation method 18‐20 .…”

Section: Introductionmentioning

confidence: 99%

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An integrable semidiscretization of the modified Camassa–Holm equation with linear dispersion term

Sheng

Feng

2022

Stud Appl Math

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In the present paper, we are with integrable discretization of a modified Camassa-Holm (mCH) equation with linear dispersion term. The key of the construction is the semidiscrete analog for a set of bilinear equations of the mCH equation. First, we show that these bilinear equations and their determinant solutions either in Gramtype or Casorati-type can be reduced from the discrete Kadomtsev-Petviashvili (KP) equation through Miwa transformation. Then, by scrutinizing the reduction process, we obtain a set of semidiscrete bilinear equations and their general soliton solution in Gram-type or Casorati-type determinant form. Finally, by defining dependent variables and discrete hodograph transformations, we are able to derive an integrable semidiscrete analog of the mCH equation. It is also shown that the semidiscrete mCH equation converges to the continuous one in the continuum limit.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An integrable semidiscretization of the modified Camassa–Holm equation with linear dispersion term

Sheng

Feng

2022

Stud Appl Math

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“…The mCH equation (1) has attracted considerable attention in the past two decades due to its rich mathematical structure and solutions. Several groups have studied the mCH equation from various aspects such as the well-posedness, regularization, Cauchy problem, Riemann-Hilbert problem, long-time asymptotics and Liouville correspondence with the mKdV equation [6][7][8][9][10][11][12][13][14] . The smooth soliton solutions were constructed by different approaches such as the Hirota's bilinear method [15,16] , Darboux transformation/Bäcklund transformation method [17][18][19] .…”

Section: Introductionmentioning

confidence: 99%

An integrable semi-discretization of the modified Camassa-Holm equation with linear dispersion term

Sheng¹,

Yu²,

Feng³

2021

Preprint

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In the present paper, we are concerned with integrable discretization of a modified Camassa-Holm equation with linear dispersion term. The key of the construction is the semi-discrete analogue for a set of bilinear equations of the modified Camassa-Holm equation. Firstly, we show that these bilinear equations and their determinant solutions either in Gram-type or Casorati-type can be reduced from the discrete KP equation through Miwa transformation. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solution in Gram-type or Casorati-type determinant form. Finally, by defining dependent variables and discrete hodograph transformations, we are able to derive an integrable semi-discrete analogue of the modified Camassa-Holm equation. It is also shown that the semi-discrete modified Camassa-Holm equation converges to the continuous one in the continuum limit.

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“…More related work on Camassa-Holm equations and short pulse equations can be found in (see [9,8,16,28,31,29,38,41,39]).…”

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confidence: 99%

Qualitative analysis for a new generalized 2-component Camassa-Holm system

Zhou¹,

Zheng²

2021

DCDS-S

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This paper considers the Cauchy problem for a 2-component Camassa-Holm system<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} m_t = ( u m)_x+ u _xm- v m, \ \ n_t = ( u n)_x+ u _xn+ v n, \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ n+m = \frac{1}{2}( u _{xx}-4 u ) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n-m = v _x $\end{document}</tex-math></inline-formula>, this model was proposed in [<xref ref-type="bibr" rid="b2">2</xref>] from a novel method to the Euler-Bernoulli Beam on the basis of an inhomogeneous matrix string problem. The local well-posedness in Sobolev spaces <inline-formula><tex-math id="M3">\begin{document}$ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ s>\frac{5}{2} $\end{document}</tex-math></inline-formula> of this system was investigated through the Kato's theory, then the blow-up criterion for this system was described by the technique on energy methods. Finally, we established the analyticity in both time and space variables of the solutions for this system with a given analytic initial data.

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On the Cauchy problem for a modified Camassa–Holm equation

Cited by 8 publications

References 25 publications

An integrable semidiscretization of the modified Camassa–Holm equation with linear dispersion term

An integrable semidiscretization of the modified Camassa–Holm equation with linear dispersion term

An integrable semi-discretization of the modified Camassa-Holm equation with linear dispersion term

Qualitative analysis for a new generalized 2-component Camassa-Holm system

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