Inverse scattering transform for new mixed spectral Ablowitz-Kaup-Newell-Segur equations

Abstract

The inverse scattering transform plays a very important role in promoting the development of analytical methods to solve non-linear PDE exactly. In this paper, new and more general mixed spectral Ablowitz-Kaup-Newell-Segur equations are derived and solved by embedding a novel time-varying spectral parameter in-to an associated linear problem and the inverse scattering transform. As a result, new exact solutions and n-soliton solutions are obtained. To gain more insights into the embedded spectral parameter and the obtained solutions, some dynamical evolutions, and spatial structures are simulated. It is shown that the derived Ablowitz-Kaup-Newell-Segur equations are Lax integrable and the obtained soliton solutions possess time-varying amplitudes.

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

“…Figures 2, 3 and 6-9 are obtained. Compared with the mixed spectral AKNS hierarchy [7] mentioned earlier in Equation (28) and the other results in [15][16][17][18], the work of this paper has some differences. Specifically, Equation (11) or its operator form (25) are different from the following equations [15][16][17][18]:…”

“…where b i and c i are all undetermined functions of x and t. Substituting Equation ( 16) into Equation (15) and comparing the coefficients of the same powers of 2ik yields:…”