Variable-Coefficient Fractional High-Order Nonlinear Models: Establishment and Solutions

This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonlinear models. Firstly, based on the matrix spectral problem proposed by Ablowitz, Kaup, Newell, and Segur, this article derives an integer-order integrable system, which is abbreviated as the AKNS hierarchy. Secondly, by taking specific values of the operator in the derived AKNS hierarchy, a variable-coefficient fractional higher-order NLS hierarchy (vfhNLSH) is obtained, and its anomalous dispersion relation (ADR) is derived via formal solution. Significantly, the reductions of the vfhNLSH include three variable-coefficient fractional-order integrable models: the Hirota equation (vfHE), the Lakshmanan–Porsezian–Daniel equation (vfLPDE), and the fifth-order NLS equation (vffNLSE). Finally, we conduct a detailed study on the representative vfHE as an example rather than a special case and construct its explicit N-fold analytical solution based on the extension of the RHA. At the same time, numerical visualization simulations are conducted to demonstrate the waveform structure characteristics of the solutions under <!-- MathType@Translator@5@5@MathML2 (no namespace).tdl@MathML 2.0 (no namespace)@ --> <math> <mrow> <mi>N</mi><mo>=</mo><mn>1</mn></mrow> </math> <!-- MathType@End@5@5@ --> and <!-- MathType@Translator@5@5@MathML2 (no namespace).tdl@MathML 2.0 (no namespace)@ --> <math> <mrow> <mi>N</mi><mo>=</mo><mn>2</mn></mrow> </math> <!-- MathType@End@5@5@ --> conditions, including solitons, breathers, and their coupled nonlinear waves. The same process is fully applicable to the other two reduced models, with only some differences in the related results and the dynamic behavior of the solutions. It is shown that the temporal part of the Lax pair associated with the vfHE cannot yet be explicitly determined. Therefore, the fractional-order extension of the RHA presented in this article constitutes a formal or RHA-inspired construction, rather than a fully rigorous fractional-order RHA extension.