论文标题
与对称性的哈密顿PDE中的周期性多脉冲和光谱稳定性
Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetry
论文作者
论文摘要
我们考虑在哈密顿系统中周期性多脉冲溶液的存在和光谱稳定性，这些溶液是不变和可逆的，对此，五阶Korteweg-de vries方程是一个原型示例。我们使用Lin的方法在周期域上构建多脉冲，特别是为周期性双脉冲展示了干草叉分叉结构。我们还使用LIN的方法来减少定期多脉冲的光谱问题来计算块矩阵的决定因素，该矩阵的决定因素编码了由相邻脉冲和与必需光谱相关的邻近脉冲和特征值之间的相互作用而产生的两个特征值。然后，我们使用此矩阵来计算与周期性单脉冲和双重脉冲相关的频谱。最值得注意的是，当特征值碰撞时，随着周期域大小的变化，我们证明了短暂的不稳定性气泡形成。这些分析结果与数值计算非常吻合，数值时间播放实验表明，这些不稳定性气泡对应于振荡不稳定性。
We consider the existence and spectral stability of periodic multi-pulse solutions in Hamiltonian systems which are translation invariant and reversible, for which the fifth-order Korteweg-de Vries equation is a prototypical example. We use Lin's method to construct multi-pulses on a periodic domain, and in particular demonstrate a pitchfork bifurcation structure for periodic double pulses. We also use Lin's method to reduce the spectral problem for periodic multi-pulses to computing the determinant of a block matrix, which encodes both eigenvalues resulting from interactions between neighboring pulses and eigenvalues associated with the essential spectrum. We then use this matrix to compute the spectrum associated with periodic single and double pulses. Most notably, we prove that brief instability bubbles form when eigenvalues collide on the imaginary axis as the periodic domain size is altered. These analytical results are all in good agreement with numerical computations, and numerical timestepping experiments demonstrate that these instability bubbles correspond to oscillatory instabilities.
