Integrable peakon equations with cubic nonlinearity
Integrable peakon equations with cubic nonlinearity
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DOI：
10.1088/1751-8113/41/37/372002
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发表时间：
2008-05
期刊：
影响因子：
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通讯作者：
A. Hone;Jing Ping Wang
中科院分区：
文献类型：
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作者：
A. Hone;Jing Ping Wang
给出了弗拉基米尔诺维科夫发现的一个新的可积偏微分方程。像Camassa-Holm和Degasperis-Procesi方程一样，这个新方程允许峰值孤子（peakon）解，但它有三次非线性项，而不是二次。我们给V Novikov方程的矩阵Lax对，并显示它是如何相关的一个倒数变换到一个负流的泽田小寺层次。不可避免地发现了许多守恒量，以及一个双哈密顿结构。后者用于获得N个峰子相互作用的有限维系统的哈密顿形式，并显式积分两体动力学（N = 2）。最后，与乔志军导出的另一个三次峰子方程的类似结果进行了比较。
We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa–Holm and Degasperis–Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada–Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of N peakons, and the two-body dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.