Riemann–Hilbert problem with arbitrary order poles for the vector Lakshmanan–Porsezian–Daniel equation

This paper investigates the vector Lakshmanan-Porsezian-Daniel (LPD) equation via the framework of the Riemann–Hilbert problem with arbitrary high-order poles. Under zero boundary conditions, we perform spectral analysis and construct the associated Riemann–Hilbert problem based on the Lax pair of the vector LPD equation. The analytical and asymptotic properties of the Jost solutions and scattering coefficients are established. By assuming that the scattering coefficients possess an N–th order zero, we derive explicit trace formulae. In the reflection-less case, we formulate and solve the general Riemann–Hilbert problem, accounting for higher-order residue conditions. This process yields explicit reconstruction formulae for the N–th order solutions of the vector LPD equation. As an example, detailed second- and third-order pole solutions are derived from the general formulae, and their asymptotic behaviors for large time are analyzed. The results demonstrate the effectiveness of the Riemann–Hilbert approach in deriving high-order localized solutions for the integrable vector LPD system.