Riemann–Hilbert Problems and Soliton Solutions of Type (??, ???) Reduced Nonlocal Integrable mKdV Hierarchies

Reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies are presented via taking two transpose-type group reductions in the Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems. One reduction is local, which replaces parameter ? with its complex conjugate ??, and other one nonlocal, negative ???. Riemann–Hilbert problems thus inverse scattering transforms formulated from reduced In view of specific distribution eigenvalues adjoint eigenvalues, soliton solutions constructed reflectionless