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.
Highlights
Starting from matrix spectral problems, one can generate integrable hierarchies of equations, based on the corresponding zero curvature equations
To present a general formulation of solutions to reflectionless Riemann–Hilbert problems in the nonlocal case, we assume that for N = 2N1 + N2, where N1, N2 ≥ 0 are two integers, we can make the rearrangements of eigenvalues λk, 1 ≤ k ≤ N and adjoint eigenvalues λk, 1 ≤ k ≤ N: λk, 1 ≤ k ≤ N : λ1, · · ·, λ N1, λ N1 +1, · · ·, λ2N1, λ2N1 +1, · · ·, λ N ∈ C+, (125)
We have proposed type (λ∗, −λ∗ ) reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies of equations, by taking advantage of two group reductions of the matrix Ablowitz–Kaup–Newell– Segur (AKNS)
Summary
Starting from matrix spectral problems, one can generate integrable hierarchies of equations, based on the corresponding zero curvature equations. Exact solutions to the resulting Riemann–Hilbert problem (9) present the required generalized matrix Jost solutions to recover the potential of the matrix spectral problems, and solutions to the corresponding integrable Equation (3). Such solutions, G + and G − , can be determined through an application of the Sokhotski–Plemelj formula to the difference of. (11) for the matrix Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems simultaneously, to generate reduced nonlocal matrix integrable mKdV hierarchies and to establish their Riemann–Hilbert problems and inverse scattering transforms.
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