Abstract
This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated Riemann-Hilbert problems. A formulation of solutions to specific Riemann-Hilbert problems, with the jump matrix being the identity matrix, is established, where eigenvalues could equal adjoint eigenvalues, and thus N-soliton solutions to the nonlocal complex reverse-spacetime mKdV hierarchies are obtained from the reflectionless transforms.
Highlights
Nonlocal integrable equations have been shown to be significantly important in modern physics, and have already become one of the hottest topics in the field of integrable equations
The inverse scattering transforms to nonlocal integrable equations have been recently developed for the scalar case [2,3,4,5,6,7] and the multicomponent case [8], and soliton solutions have been generated from the Riemann-Hilbert problems with the identity jump matrix [8,9], via Darboux transformations [10,11,12,13,14] and the Hirota bilinear method [15,16]
Based on the Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem with multiple potentials, we would like to construct a class of modified Korteweg-de Vries (mKdV) hierarchies of nonlocal complex reverse-spacetime integrable equations and to discuss their inverse scattering transforms and soliton solutions from a perspective of Riemann-Hilbert problems
Summary
Nonlocal integrable equations have been shown to be significantly important in modern physics, and have already become one of the hottest topics in the field of integrable equations. Based on the Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem with multiple potentials, we would like to construct a class of mKdV hierarchies of nonlocal complex reverse-spacetime integrable equations and to discuss their inverse scattering transforms and soliton solutions from a perspective of Riemann-Hilbert problems. Ξ−λ if G has no pole, and the resulting solutions can be used to build the needed generalized matrix Jost functions for retrieving the potential in the original matrix spectral problems, which solves the given integrable equation. Through such a formulation of solutions, we compute soliton solutions for the nonlocal complex reverse-spacetime mKdV hierarchies. We present a conclusion, along with a few futuristic remarks
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