New Integrable Multi-L\'evy-Index and Mixed Fractional Nonlinear Soliton Hierarchies

TL;DR

This paper introduces new integrable hierarchies involving multi-Lévy-index and mixed fractional nonlinear equations, providing explicit forms, dispersion relations, and multi-soliton solutions to advance understanding of nonlinear wave transport in fractional media.

Contribution

It presents a novel method to generate and explicitly formulate new integrable fractional soliton hierarchies with multi- and mixed-Lévy indices, including their solutions and dispersion relations.

Findings

Derived explicit forms of fractional hierarchies.

Obtained fractional multi-soliton solutions.

Analyzed anomalous dispersion relations.

Abstract

In this letter, we present a simple and new idea to generate two types of novel integrable multi-L\'evy-index and mix-L\'evy-index (mixed) fractional nonlinear soliton hierarchies, containing multi-index and mixed fractional higher-order nonlinear Schr\"odinger (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.