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

Abstract

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