ON THE VARIATIONAL STRUCTURE OF BREATHER SOLUTIONS II: PERIODIC MKDV EQUATION
ON THE VARIATIONAL STRUCTURE OF BREATHER SOLUTIONS II: PERIODIC MKDV EQUATION
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呼吸器解的变分结构II：周期MKDV方程
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
J. M. Palacios
中科院分区：
文献类型：
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作者：
Miguel A. Alejo;Claudio Muñoz;J. M. Palacios
我们研究了周期修正的KdV方程，其中从Kevrekidis等人的工作中已知空间和时间上的周期呼吸子解。我们证明了这些呼吸子满足一个适当的椭圆方程，并通过数值方法讨论了其谱稳定性。我们还确定了[19]中描述的情况下的非线性不稳定性的来源，并且我们推测，即使满足谱稳定性，非线性稳定性/不稳定性仅取决于合适的判别函数的符号，在非周期（空间）mKdV呼吸子的情况下，该条件是平凡满足的。最后，我们提出了一类新的呼吸mKdV的解决方案，相信存在的几何考虑，这是周期性的时间和空间，但有非零的平均值，不像标准呼吸。
We study the periodic modified KdV equation, where a periodic in space and time breather solution is known from the work of Kevrekidis et al. [19]. We show that these breathers satisfy a suitable elliptic equation, and we also discuss via numerics its spectral stability. We also identify a source of nonlinear instability for the case described in [19], and we conjecture that, even if spectral stability is satisfied, nonlinear stability/instability depends only on the sign of a suitable discriminant function, a condition that is trivially satisfied in the case of non-periodic (in space) mKdV breathers. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.