Inverse scattering transform of the focusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

This work applies the inverse scattering transform approach to investigate the initial value problem of the focusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions (i.e., when the asymptotic phases and amplitudes are asymmetric at spatial infinity). In the context of the direct problem, the analyticity properties and symmetry relations of the Jost solutions and scattering coefficients are thoroughly explored without introducing a uniformization variable, and their asymptotic behavior as the scattering parameter tends to infinity is derived. Furthermore, the inverse problem is formulated using the Marchenko integral equations and the matrix Riemann–Hilbert problem on the single sheet of the scattering variables. Finally, the time evolutions of the scattering coefficients and eigenfunctions are constructed, demonstrating their nontrivial dependence on time.