Interactions of fractional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e2903" altimg="si3.svg"><mml:mi>N</mml:mi></mml:math>-solitons with anomalous dispersions for the integrable combined fractional higher-order mKdV hierarchy

In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann–Hilbert (RH) problem, and fractional multi-solitons of integrable combined higher-order mKdV (fhmKdV) hierarchy, including (fmKdV), fifth-order (f5mKdV), third–fifth-order (f35mKdV) equations, etc., which can be featured via completeness squared scalar eigenfunctions ZS spectral problem. We construct matrix RH problem to present three types N-solitons illustrating dispersions fhmKdV hierarchy for reflectionless case. As some examples, analyse wave velocity one-soliton such that find equation predicts power law relationship between amplitude, demonstrates dispersion. Furthermore, illustrate other interesting phenomena containing elastic interactions bright dark solitons, W-shaped soliton soliton, as well breather soliton. These obtained will useful understand related nonlinear super-dispersive propagations in media.