Global paths of time-periodic solutions of the Benjamin–Ono equation connecting pairs of traveling waves
Global paths of time-periodic solutions of the Benjamin–Ono equation connecting pairs of traveling waves
复制标题
连接行波对的本杰明-小野方程时间周期解的全局路径
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
J. Wilkening
中科院分区：
文献类型：
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作者：
David M. Ambrose;J. Wilkening
我们分类所有的分支，从行波到非平凡的时间周期解的Benjamin-Ono方程的线性化预测。我们使用一个谱精确的数值延拓方法来研究几个路径的非平凡的解决方案超出了线性理论的领域。发现这些路径要么与不同的行波重新连接，要么爆炸。在后一种情况下，当分叉参数接近临界值时，初始条件的振幅无限制地增长，周期接近零。然后我们证明了一个定理，给出了从一个分支映射到其对应的另一边的路径，并展示了时间周期解的精确公式，这些解决方案的Fourier系数是有限数量的粒子位置的幂和，其基本对称功能执行简单的轨道（圆或周转圆）在单位圆盘的复平面。我们也发现这些路径已经非平凡的解决方案的内部分叉的例子，但我们不试图分析其解析结构。
We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We then prove a theorem that gives the mapping from one bifurcation to its counterpart on the other side of the path and exhibits exact formulas for the time-periodic solutions on this path. The Fourier coefficients of these solutions are power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits (circles or epicycles) in the unit disk of the complex plane. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their analytic structure.