可积分数阶高阶非线性 Schrödinger 方程反常色散的分数阶 N-孤子解动力学
Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schrödinger equations
AI总结：
该研究利用含简单极点的矩阵 Riemann-Hilbert 逆散射方法，求解可积 fHONLS 方程在无反射情形下的分数阶 N-孤子，并揭示 fHirota 与 fcmKdV 孤子的反常色散及速度—振幅幂律关系。
摘要：
本文研究可积分数阶高阶非线性 Schrödinger（fHONLS）方程的反常色散关系、逆散射变换和分数阶 N-孤子解，这些方程包括分数阶 Hirota（fHirota）、分数阶复 mKdV（fcmKdV）以及分数阶 Lakshmanan-Porsezian-Daniel（fLPD）方程等。借助具有简单极点的矩阵 Riemann-Hilbert 问题，可以精确求解逆散射问题。由此，在无反射情形下得到了 fHONLS 方程分数阶 N-孤子解的显式公式。特别地，作者分析了 fHirota 和 fcmKdV 方程具有反常色散的分数阶单孤子、双孤子和三孤子解。这些包络分数阶单孤子解的波速、群速度和相速度与其振幅的幂律相关。所得分数阶 N-孤子解或有助于解释分数阶非线性介质中非线性波的相关超色散输运现象。
英文摘要：
In this paper, we explore the anomalous dispersive relations, inverse scattering transform and fractional N-soliton solutions of the integrable fractional higher-order nonlinear Schrodinger (fHONLS) equations, containing the fractional Hirota (fHirota), fractional complex mKdV (fcmKdV), and fractional Lakshmanan-Porsezian-Daniel (fLPD) 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 fHirota and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional 1-soliton solutions are related to the power laws of their amplitudes. These obtained fractional N-soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media.
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