An Integrable Evolution System and Its Analytical Solutions with the Help of Mixed Spectral AKNS Matrix Problem

Abstract

In this work, a novel integrable evolution system in the sense of Lax’s scheme associated with a mixed spectral Ablowitz-Kaup-Newell-Segur (AKNS) matrix problem is first derived. Then, the time dependences of scattering data corresponding to the mixed spectral AKNS matrix problem are given in the inverse scattering analysis. Based on the given time dependences of scattering data, the reconstruction of potentials is carried out, and finally analytical solutions with four arbitrary functions of the derived integrable evolution system are formulated. This study shows that some other systems of integrable evolution equations under the resolvable framework of the inverse scattering method with mixed spectral parameters can be constructed by embedding different spectral parameters and time-varying coefficient functions to the known AKNS matrix spectral problem.

Keywords

• integrable evolution system
• mixed spectral AKNS matrix problem
• spectral parameter
• scattering data
• analytical solution
• inverse scattering method