Integrable fractional modified Korteweg–deVries, sine-Gordon, and sinh-Gordon equations
Integrable fractional modified Korteweg–deVries, sine-Gordon, and sinh-Gordon equations
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DOI：
10.1088/1751-8121/ac8844
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
M. Ablowitz;Joel B. Been;L. Carr
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文献类型：
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作者：
M. Ablowitz;Joel B. Been;L. Carr
逆散射变换允许显式构造许多物理上重要的非线性波动方程的解。值得注意的是，利用相关线性散射问题的合适特征函数的完备性，该方法可以推广到以反常色散为特征的分数阶非线性演化方程。在反常扩散中，均方位移与t α， α > 0成正比，而在反常色散中，局域波的速度与A α成正比，其中A为波的振幅。得到了改进的Korteweg-deVries （mKdV）、sin - gordon （sing）和sinh-Gordon （sinhG）及其相关层次的分数扩展。利用线性散射问题中存在的对称性，这些方程可以与分数阶mKdV （fmKdV）、分数阶sing （fsing）和分数阶sinhG （fsinhG）为特殊情况的非线性演化方程标量族联系起来。得到了标量问题解的完备性，并由此得到了用谱展开表示的非线性演化方程。特别地，fmKdV、fsineG和fsinhG是显式编写的。利用逆散射变换导出了fmKdV和fsineG的单孤子解，并证明了这些孤子具有异常色散。
The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A α , where A is the amplitude of the wave. Fractional extensions of the modified Korteweg–deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion.