MATLAB_code_RSPA-2024-0993.zip from Instability bands for periodic travelling waves in the modified Korteweg–de Vries equation

Two families of periodic travelling waves exist in the focusing modified Korteweg–de Vries equation. Spectral stability of these waveforms with respect to co-periodic perturbations of the same period has been previously explored by using spectral analysis and variational formulation. By using tools of integrability, such as a relation between squared eigenfunctions of the Lax pair and eigenfunctions of the linearized stability problem, we revisit the spectral stability of these waveforms with respect to perturbations of arbitrary periods. In agreement with previous works, we find that one family is spectrally stable for all parameter configurations, whereas the other family is spectrally unstable for all parameter configurations. We show that the onset of the co-periodic instability for the latter family changes the instability bands from figure-8 (crossing at the imaginary axis) into figure-∞ (crossing at the real axis).

在聚焦型修正Korteweg-de Vries（focusing modified Korteweg–de Vries）方程中，存在两类周期行波族。此前已有研究通过谱分析与变分形式，针对这类波形在同周期扰动下的谱稳定性开展了研究。本文借助可积性相关工具——例如Lax对（Lax pair）的平方本征函数与线性化稳定性问题本征函数之间的关联——重新审视了这类波形在任意周期扰动下的谱稳定性。与既往研究结果一致，我们发现其中一类波族在所有参数配置下均保持谱稳定，而另一类波族在所有参数配置下均呈现谱不稳定性。我们还证明，针对后者的同周期不稳定性起始点，会将其不稳定带从“8字形（在虚轴处相交）”转变为“∞字形（在实轴处相交）”。