Waltzing peakons and compacton pairs in a cross-coupled Camassa–Holm equation
Waltzing peakons and compacton pairs in a cross-coupled Camassa–Holm equation
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DOI：
10.1088/1751-8113/44/26/265205
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发表时间：
2011-03
期刊：
影响因子：
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通讯作者：
Colin J Cotter;Darryl D. Holm;R. Ivanov;J. Percival
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文献类型：
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作者：
Colin J Cotter;Darryl D. Holm;R. Ivanov;J. Percival
研究了两个交叉耦合的Camassa-Holm(CCCH)方程组的奇性解。这个CCCH系统允许峰解，但它不在二分量CH可积族中。该系统是一对关于实线上两类解的耦合哈密顿偏微分方程组，每一类方程分别具有EXP(−|x|)峰解，其中一阶导数在峰处不连续。然而，由于不存在自相互作用，所以两种类型的峰子解都只在另一种类型的诱导速度下运动。分析了单束缚峰偶(峰偶)情况下的“华尔兹”解的行为，以及峰偶的超越碰撞和峰偶与反峰偶迎头碰撞的反对称情形。然后我们给出了这些碰撞的数值解，它们是非弹性的，因为每个华尔兹峰偶都有一个内部自由度对应于它们的‘节奏’--也就是在相空间中相反类型的两个峰子在相空间中相互循环的周期。最后，我们讨论了交叉耦合Euler-Poincaré(CCEP)方程的紧致偶解，并说明了与尖峰偶、三角形紧致偶和抛物型紧致偶解相同的碰撞类型。最后，我们提出了一些悬而未决的问题和挑战，以便理解CCCH和CCEP方程的耦合动力学。
We consider singular solutions of a system of two cross-coupled Camassa–Holm (CCCH) equations. This CCCH system admits peakon solutions, but it is not in the two-component CH integrable hierarchy. The system is a pair of coupled Hamiltonian partial differential equations for two types of solutions on the real line, each of which separately possesses exp ( − |x|) peakon solutions with a discontinuity in the first derivative at the peak. However, there are no self-interactions, so each of the two types of peakon solutions moves only under the induced velocity of the other type. We analyse the ‘waltzing’ solution behaviour of the cases with a single bound peakon pair (a peakon couple), as well as the over-taking collisions of peakon couples and the antisymmetric case of the head-on collision of a peakon couple and a peakon anti-couple. We then present numerical solutions of these collisions, which are inelastic because the waltzing peakon couples each possess an internal degree of freedom corresponding to their ‘tempo’—that is, the period at which the two peakons of opposite type in the couple cycle around each other in phase space. Finally, we discuss compacton couple solutions of the cross-coupled Euler–Poincaré (CCEP) equations and illustrate the same types of collisions as for peakon couples, with triangular and parabolic compacton couples. We finish with a number of outstanding questions and challenges remaining for understanding couple dynamics of the CCCH and CCEP equations.