论文标题
可集成的分数修改Korteweg-de Vries，Sine-Gordon和Sinh-Gordon方程
Integrable Fractional Modified Korteweg-de Vries, Sine-Gordon, and Sinh-Gordon Equations
论文作者
论文摘要
反向散射变换允许明确构建溶液，以实现许多物理上重要的非线性波方程。值得注意的是，该方法可以扩展到以异常分散为特征的分数非线性进化方程，该方程是使用相关线性散射问题的合适本征函数的完整性。在异常的扩散中，平均平方位移与$ t^α$，$α> 0 $成正比，而在异常分散方面，局部波的速度与$ a^α$成比例，其中$ a $ a $是波浪的幅度。获得了修改后的Korteweg-Devries（MKDV），Sine-Gordon（Sineg）和Sinh-Gordon（SINHG）的分数扩展以及相关的层次结构。使用线性散射问题中存在的对称性，可以将这些方程与非线性进化方程的标量家族相连，该方程是特殊情况。获得了标量问题的解决方案的完整性，从中，非线性进化方程是根据光谱膨胀的特征。特别是，FMKDV，FSINEG和FSINHG是明确编写的。使用反向散射变换得出了FMKDV和FSINEG的一氧化溶液，并且这些孤子被证明表现出异常的分散体。
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.
