New integrable multi-Lévy-index and mixed fractional nonlinear soliton hierarchies

In this letter, we present a simple and new idea to generate two types of novel integrable multi-Lévy-index and mix-Lévy-index (mixed) fractional nonlinear soliton hierarchies, containing multi-index and mixed fractional higher-order nonlinear Schrödinger (NLS) hierarchy, fractional complex modified Korteweg–de Vries (cmKdV) hierarchy, and fractional mKdV hierarchy. Their explicit forms can be given using the completeness of squared eigenfunctions. Moreover, we present their anomalous dispersion relations via their linearizations, and fractional multi-soliton solutions via the inverse scattering transform with matrix Riemann–Hilbert problems. These obtained fractional multi-soliton solutions may be useful to understand the related super-dispersion transports of nonlinear waves in multi-index fractional nonlinear media.