Riemann–Hilbert Approach and Multiple Arbitrary-Order Pole Solutions for the Lakshmanan–Porsezian–Daniel Equation with Finite Density Initial Data

In this work, the Riemann–Hilbert (RH) problem is employed to study the Lakshmanan–Porsezian–Daniel (LPD) equation with arbitrary-order pole points under finite density initial data condition. By performing spectral analysis on Lax pairs, a suitable matrix RH problem is established. Through the residue theorem, the explicit expression of simple pole solutions is obtained by Binet–Cauchy theorem. In addition, utilizing the Wronskian form of scattering data \(s_{11}(\mu )\) which degenerates to zero at high-order zero points and the Taylor expansion of oscillation index \(e^{2i\theta }\), the expression of the high-order pole solutions is constructed. Moreover, the detailed analysis is conducted on the dynamic behaviors of special soliton solutions, and some interesting soliton phenomena are presented by taking the influence of various parameters into consideration.