Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
Chaos. 2022 Dec;32(12):123110. doi: 10.1063/5.0101921.
In this paper, using the algorithm due to Ablowitz et al. [Phys. Rev. Lett. 128, 184101 (2022); J. Phys. A: Math. Gen. 55, 384010 (2022)], we explore the anomalous dispersive relations, inverse scattering transform, and fractional N-soliton solutions of the integrable fractional higher-order nonlinear Schrödinger (fHONLS) equations, containing the fractional third-order NLS (fTONLS), fractional complex mKdV (fcmKdV), and fractional fourth-order nonlinear Schrödinger (fFONLS) equations, etc. The inverse scattering problem can be solved exactly by means of the matrix Riemann-Hilbert problem with simple poles. As a consequence, an explicit formula is found for the fractional N-soliton solutions of the fHONLS equations in the reflectionless case. In particular, we analyze the fractional one-, two-, and three-soliton solutions with anomalous dispersions of fTONLS and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional one-soliton solutions are related to the power laws of their amplitudes. Moreover, we also deduce the formula for the fractional N-soliton solutions of all fHONLS equations and analyze some velocities of the one-soliton solution. These obtained fractional N-soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media.
在本文中，我们使用 Ablowitz 等人的算法[Phys. Rev. Lett. 128, 184101 (2022); J. Phys. A: Math. Gen. 55, 384010 (2022)]，研究了可积分数阶高阶非线性薛定谔（fHONLS）方程的反常色散关系、逆散射变换和分数 N 孤子解，包含分数三阶非线性薛定谔（fTONLS）、分数复 mKdV（fcmKdV）和分数四阶非线性薛定谔（fFONLS）等方程。逆散射问题可以通过具有简单极点的矩阵黎曼-希尔伯特问题精确求解。因此，在无反射情况下找到了 fHONLS 方程的分数 N 孤子解的显式公式。特别是，我们分析了 fTONLS 和 fcmKdV 方程的反常色散的分数一、二和三孤子解。这些包络分数一孤子解的波、群和相位速度与它们振幅的幂律有关。此外，我们还推导出了所有 fHONLS 方程的分数 N 孤子解的公式，并分析了一孤子解的一些速度。这些得到的分数 N 孤子解可能有助于解释分数非线性介质中非线性波的相关超色散传输。
