General multi-soliton and higher-order soliton solutions for a novel nonlocal Lakshmanan–Porsezian–Daniel equation

Abstract

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, a novel integrable nonlocal LPD equation corresponding to a \(3\times 3\) Lax pair is proposed. Secondly, the inverse scattering process with a novel left-right \(3\times 3\) matrix Riemann–Hilbert(RH) problem is constructed. The analytical properties and symmetry relations for the Jost functions and scattering data are considerably different from the local ones. Due to the special symmetry properties for the nonlocal LPD equation, the zeros of the RHP problem are purely imaginary or occur in pairs. With different types and configuration of zeros, the soliton formula is provided and the rich dynamical behaviors for the three kinds of multi-solitons for the novel nonlocal LPD equation are demonstrated. Third, by a technique of adding perturbed parameters and limiting process, the formula of higher-order solitons for the nonlocal LPD equation is exhibited. Lastly, the plots of diverse higher-order solitons and various solutions corresponding to different combinations of the following zeros: purely imaginary higher-order zeros, purely imaginary simple zeros, pairs of non-purely imaginary simple zeros and pairs of non-purely imaginary higher-order zeros are displayed.