Riemann-Hilbert Problems and Soliton Solutions of Type (λ*, -λ*) 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 matrix Ablowitz-Kaup-Newell-Segur (AKNS) spectral problems. One reduction is local, which replaces the spectral parameter lambda with its complex conjugate lambda(*), and the other one is nonlocal, which replaces the spectral parameter lambda with its negative complex conjugate -lambda(*). 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.

Automated summary

Reduced nonlocal matrix integrable modified Korteweg-de Vries (mKdV) hierarchies are obtained through two transpose-type group reductions in the matrix Ablowitz-Kaup-Newell-Segur (AKNS) spectral problems. Riemann-Hilbert problems and soliton solutions are formulated based on the reduced matrix spectral problems.