Analytical and numerical studies for integrable and non-integrable fractional discrete modified Korteweg-de Vries hierarchies.

Under investigation in this paper is the integrable and non-integrable fractional discrete modified Korteweg-de Vries hierarchies. The linear dispersion relations, completeness relations, inverse scattering transform, and fractional soliton solutions of the integrable fractional discrete modified Korteweg-de Vries hierarchy will be explored. The inverse scattering problem will be solved accurately by constructing Gel'fand-Levitan-Marchenko equations and Riemann-Hilbert problem. The peak velocity of fractional soliton solutions will be analyzed. Numerical solutions of the non-integrable fractional averaged discrete modified Korteweg-de Vries equation, which has a simpler form than the integrable one, will be obtained by a split-step Fourier scheme.