Exact solutions and their local structures of an inverse scattering integrable system with variable coefficients

An important manifestation of the integrability of nonlinear mathematical and physical models is their solvability through inverse scattering transform (IST). The problem to be investigated in this paper is a system of fifth-order nonlinear evolution equations (NLEEs) with variable coefficients related to time-varying spectral problems. The expected result is to derive the system of fifth-order NLEEs, obtain its exact solutions, verify its integrability in the sense IST solvability, and reveal some novel local structures of the obtained solutions. First, the system of fifth-order NLEEs is derived. Then, by combining the IST with a time-varying spectral parameter, the associated scattering data are determined for the reconstruction of potentials. Using the determined scattering data, exact solutions are ultimately obtained. Meanwhile, some local structures with new features are analyzed for two pairs of special solutions, including kink solitons, wide/narrow top bell solitons, and small-scale peaks in the time variable direction, as well as sine/cosine-like fluctuations in the spatial variable direction. This paper not only demonstrates that equipping the Ablowitz–Kaup–Newell–Segur (AKNS) problems with appropriate time-varying spectra can be used to derive some other inverse scattering integrable systems of NLEEs with variable coefficients, but also graphically illustrates the regulatory effect of time-varying spectral parameter on local solution structures.