网络中的非线性波：一种使用sine-Gordon方程的简单方法
Nonlinear waves in networks: a simple approach using the sine-Gordon equation
摘要：
为了研究非线性波在Y型和T型结中的传播，我们以二维sine-Gordon方程为模型，研究了这种几何结构中扭结和呼吸子的动力学行为。能量的比较表明，分叉的角度不起作用。受此启发，我们引入了一个一维的有效方程，其解决方案同意以及与2D模拟扭结和呼吸器的解决方案。对于宽度相等的分支，呼吸器交叉大约发生在$v > 1 - ω$时，其中$v$是呼吸器蜂窝，$ω$是其频率。然后，我们通过估计它们的速度和频率来描述两个上分支中的呼吸。这些新的呼吸器比最初的呼吸器慢，并且频率上移。从理论上讲，本研究可以推广到更复杂的非线性波。
英文摘要：
To study the propagation of nonlinear waves across Y- and T-type junctions, we consider the 2D sine--Gordon equation as a model and study the dynamics of kinks and breathers in such geometries. The comparison of the energies reveals that the angle of the fork plays no role. Motivated by this, we introduce a 1D effective equation whose solutions agree well with the 2D simulations for kink and breather solutions. For branches of equal width, breather crossing occurs approximately when $v > 1 - ω$, where $v$ is the breather celerity and $ω$ is its frequency. We then characterize the breathers in the two upper branches by estimating their velocity and frequency. These new breathers are slower than the initial breather and up-shifted in frequency. In perspective, this study could be generalized to more complex nonlinear waves.
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