Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity
Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity
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DOI：
10.1088/1361-6544/ac93e1
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发表时间：
2022-02
期刊：
影响因子：
1.7
通讯作者：
Isaac Friedman;Oscar G. Riaño;S. Roudenko;Diana Son;Kai Yang
中科院分区：
文献类型：
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作者：
Isaac Friedman;Oscar G. Riaño;S. Roudenko;Diana Son;Kai Yang
我们考虑两种类型的广义Korteweg-de弗里斯方程，其中的非线性给出或没有绝对值，特别是，包括低功率的非线性，其中的一个例子是Schamel方程。我们首先证明了这两个方程在H 1的加权子空间中的局部适定性，其中包括多项式衰减的函数，扩展了Linares等人（2019 Commun.同温度数学21 1850056）到分数权重。然后，我们调查解决方案的数值，确认适定性，并将其扩展到更广泛的一类功能，包括指数衰减。我们包括这两种类型的方程的解决方案的比较，特别是，我们调查孤子分辨率的正，负的数据与不同的衰减率。最后，我们研究了两种模型中各种孤立波的相互作用，显示了孤子，色散辐射甚至呼吸子的形成，所有这些都更容易在低功率的非线性下跟踪。
We consider two types of the generalized Korteweg–de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of H 1 that includes functions with polynomial decay, extending the result of Linares et al (2019 Commun. Contemp. Math. 21 1850056) to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power.