Blow-Up Solutions and Peakons to a Generalized μ-Camassa–Holm Integrable Equation
Blow-Up Solutions and Peakons to a Generalized μ-Camassa–Holm Integrable Equation
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DOI：
10.1007/s00220-014-2007-z
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发表时间：
2013-05
影响因子：
2.4
通讯作者：
C. Qu;Ying Fu;Yue Liu
中科院分区：
文献类型：
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作者：
C. Qu;Ying Fu;Yue Liu
本文考虑一个广义μ-型可积方程，它可以看作是μ-Camassa-Holm方程和修正的μ-Camassa-Holm方程的推广.结果表明，该方程在形式上是可积的，具有Lax对和双Hamilton结构，其尺度极限是描述短毛细重力波的流体动力系统的可积模型.在适当的Sobolev空间中，利用粘性方法建立了Cauchy问题的局部适定性。研究了该方程峰值行波解的存在性和解的奇性的形成。发现该方程存在单峰和多峰行波解。详细说明了不同的μ-Camassa-Holm和修正的μ-Camassa-Holm非局部非线性对爆破准则和波浪破碎的影响。我们的分析依赖于特征和守恒量的方法，并进行了先验差分估计。
Considered here is a generalizedμ-type integrable equation, which can be regarded as a generalization to both theμ-Camassa–Holm and modifiedμ-Camassa–Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling wave solutions and formation of singularities of solutions for the equation are investigated. It is found that the equation admits single and multi-peaked traveling wave solutions. The effects of varyingμ-Camassa–Holm and modifiedμ-Camassa–Holm nonlocal nonlinearities on blow-up criteria and wave breaking are illustrated in detail. Our analysis relies on the method of characteristics and conserved quantities and is proceeded with a priori differential estimates.