Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetry
Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetry
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DOI：
10.1016/j.jde.2022.06.019
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发表时间：
2020-10
影响因子：
2.4
通讯作者：
Ross Parker;Bjorn Sandstede
中科院分区：
文献类型：
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作者：
Ross Parker;Bjorn Sandstede
考虑平移不变可逆哈密顿系统周期多脉冲解的存在性和谱稳定性，其中五阶Korteweg-de Vries方程是一个典型的例子。我们利用Lin的方法构造了周期域上的多脉冲，并特别证明了周期双脉冲的干草叉分叉结构。我们还使用Lin的方法将周期多脉冲的频谱问题简化为计算块矩阵的行列式，该行列式编码由相邻脉冲之间相互作用产生的特征值和与本质频谱相关的特征值。然后我们使用这个矩阵来计算与周期性单脉冲和双脉冲相关的频谱。最值得注意的是，我们证明了当周期域大小改变时特征值在虚轴上碰撞时，会形成短暂的不稳定气泡。这些分析结果与数值计算结果吻合较好，数值时步实验表明，这些不稳定气泡对应于振荡不稳定。
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.