Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

Abstract

The Riemann–Hilbert approach is demonstrated to investigate the defocusing Lakshmanan–Porsezian–Daniel equation under fully asymmetric nonzero boundary conditions. In contrast to the symmetry case, this paper focuses on the branch points related to the scattering problem rather than using the Riemann surfaces. For the direct problem, we analyze the Jost solution of lax pairs and some properties of scattering matrix, including two kinds of symmetries. The inverse problem at branch points can be presented, corresponding to the associated Riemann–Hilbert. Moreover, we investigate the time evolution problem and estimate the value of solving the solutions by Jost function. For the inverse problem, we construct it as a Riemann–Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan–Porsezian–Daniel equation. The solutions of the Riemann–Hilbert problem can be constructed by estimating the solutions. Finally, we work out the solutions under fully asymmetric nonzero boundary conditions precisely via utilizing the Sokhotski–Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces. These results are valuable for understanding physical phenomena and developing further applications of optical problems.