AI 中文总结
该研究分析Clifford代数推广的Camassa-Holm方程的双峰子动力学，确立峰间同步能量交换的解析结果，揭示其振荡能量转移与渐近解耦的新特征。
AI 中文摘要
我们研究了由Euler-Bernoulli梁问题的重新表述产生的Camassa-Holm方程的双分量扰动的动力学，该扰动最近被推广到一般Clifford代数场景中。我们聚焦于与具有两个生成元且Minkowski符号的Clifford代数相关的原始情况，对于该情况，所得方程允许带有内部自由度的非光滑孤子解（峰子）。我们通过解析和数值方法研究了双峰子解的动力学，确立了空间分离的峰之间存在同步的能量交换，这一现象此前仅通过数值方法观测到。我们获得了长时间动力学的完整描述：振幅趋近于由谱不变量确定的周期轨道，而由此产生的隐藏周期性控制着两个峰子之间的持续交换。极限轨道及其平均动力学通过椭圆函数和完全椭圆积分得到明确描述。我们还推导了一个将峰间距与两个振幅的累积失衡相关联的精确恒等式。该恒等式结合冻结参数比较论证，得出峰子之间相互作用的指数衰减以及振幅变量向极限周期轨道的指数收敛。此外，一旦达到渐近状态，即使瞬时速率可能继续改变符号，峰间距也会从内部振荡的一个周期到下一个周期增大。这些结果揭示了标量Camassa-Holm方程中不存在的动力学特征：在不断分离的峰子之间发生持续的振荡能量转移，同时伴有定量控制的渐近解耦。
英文摘要
We study the dynamics of a two-component perturbation of the Camassa--Holm equation arising from a reformulation of the Euler--Bernoulli beam problem, recently extended to a general Clifford algebra setting. We focus on the original case associated with a Clifford algebra with two generators and Minkowski signature, for which the resulting equation admits nonsmooth soliton solutions (peakons) carrying internal degrees of freedom. We investigate analytically and numerically the dynamics of a two-peakon solution and establish the existence of a synchronized exchange of energy between spatially separated peaks, a phenomenon previously observed only numerically. We obtain a complete description of the long-time dynamics: the amplitudes approach a periodic orbit determined by the spectral invariants, and the resulting hidden periodicity governs the persistent exchange between the two peakons. The limiting orbit and its averaged dynamics are described explicitly in terms of elliptic functions and complete elliptic integrals. We also derive an exact identity relating the peak separation to the accumulated imbalance of the two amplitudes. This identity, combined with a frozen-parameter comparison argument, yields exponential decay of the interaction between the peakons and exponential convergence of the amplitude variables to the limiting periodic orbit. Moreover, once the asymptotic regime is reached, the separation increases from one period of the internal oscillation to the next, even though its instantaneous rate may continue to change sign. These results reveal a dynamical feature absent from the scalar Camassa--Holm equation: a persistent oscillatory transfer of energy between increasingly separated peakons, coupled with quantitatively controlled asymptotic decoupling.
